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THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


Music 
Library 


A 


PRIMER 


OF 


PDERN  MUSICAL  TONALITY; 


INTENDED  FOR  SELF-STUDY, 


AS   AN    INTRODUCTION    TO 


THE    STUDY    OF    H  A  E  M  O  ^  Y. 


BY 

J.    H.    CORNELL. 


THIRD     EDITION,    REVISED    AND    IMPROVED. 


NEW    YORK: 

G.    SCHIRMEF-J,   35   UNION   SQUARE, 

1904 


COPTRIGHT,  1877,  BY  G.   SCHIRMER 


Note. — The  author  of  this  little  work  has  availed  himself  of  the  oppor- 
tunity of  correcting,  in  this  third  edition,  some  slight  inaccuracies  in  the 
first,  to  which  his  latei  studies  in  Harmony  have  called  his  attention. 


Electrotyped  by  Smith  &  McDodgal,  82  Beekman  St. 


Music 
Library 


/ 


INTRODUCTION 


MUSIC,  whose  noblest  mission  it  is  to  awaken  in  the  soul  cer- 
tain sentiments  and  affections,  uses  for  this  purpose  fleeting 
sounds  or  to7ies,  whence  it  may  be  briefly  defined  as  the  Science  or 
Art  of  Tone — as,  indeed,  it  is  called  in  German  die  Tonhunst, 
literally,  "  Tone-art." 

The  explanation  of  the  physical  laws  of  the  production  of  mu- 
sical sounds,  or  tones,  belongs  to  the  Science  of  Acoustics.  Suffice 
it  to  say,  in  this  place,  that  tone  is  the  result  of  rapid  vibrations 
(oscillations)  of  some  elastic  body  in  a  state  of  alternate  tension* 
and  relaxation.  These  vibrations,  agitating  the  air,  cause  sound- 
waves, and  thus  reach  our  ear,  producing  upon  us  the  impression 
of  a  tone,  which  is  liigh  (acute),  or  low  (grave),  as  to  its  pitch,  ac- 
cording as  the  vibrations  are  very  rapid  or  less  so,  and  the  sound- 
waves in  the  air  correspondingly  short  or  long. 

The  extremes  of  pitch  recognized  in  our  musical  system  are 
represented  by  the  lowest  tone  of  the  largest  modem  Organ  and 
the  highest  tone  of  the  modern  Piano-forte.  The  lowest  tone  used 
in  the  Organ,  viz.:  C^ — found  only  in  instruments  of  the  first  class 
— is  produced  by  a  pedal-pipe  32  ft.  in  length  (whence  the  tone  is 

*  The  word  "  tone  "  comes  from  tlie  Greek  "  tonos,"  originally  signifying 
tension. 


M        -   ■»■    ATL.. 


4  INTRODUCTION. 

called  32  ft.  C),  and  has  16^  double  vibrations*  in  a  second:  it  is, 
liowever,  pretty  generally  conceded  that  this  tone  is  too  deep  to  be 
perceived  by  the  ear  as  a  tone  of  a  certain  pitch,  and  that  it  is  not 
until  we  reach  41^  -snbrations  per  second,  givuig  E^,  the  lowest 
tone  sounded  by  the  double-bass,  that  we  obtain  a  musical  tone 
proper.  The  highest  tone  of  the  modern  Piano-forte  of  full  com- 
pass, viz. :  C'g,  is  produced  by  4096  vibrations  per  second.  From 
the  lower  to  the  higher  of  these  two  extremes  there  is  a  contumous 
rise  in  pitch,  passing  through  every  possible  gradation,  thus  afford- 
ing a  multitude  of  tones  mathematically  different  f  (i.  e.,  each  tone 
having  its  own  fixed  number  of  vibrations),  from  which  multitude 
a  certain  series  of  tones,  with  no  smaller  difference  of  pitch  between 
any  two  than  that  of  a  half-step,  has  been  selected  to  constitute  the 
material  of  our  modern  musical  system. 

Tones,  as  forming  the  material  of  music,  have  three  principal 


*  By  a  dovUe  vibration  is  meant  the  complete  excursion — say  of  a  sound- 
ing string — backwards  and  forwards,  just  as  the  movement  of  a  pendulum  to 
the  right  and  back  to  the  left,  or  vice-versa,  constitutes  a  complete  or  double 
oscillation.  The  word  "  vibration  "  is  used  by  English-speaking  writers  in  the 
complete  sense  ;  by  the  French  and  Germans,  however,  generally  in  the  sense 
of  a  single  vibration. 

f  These  extremely  fine  distinctions  of  pitch  may  be  produced  to  a  certain 
extent  on  such  stringed  instruments  as  the  violin,  etc.,  but  most  perfectly  by 
the  aid  of  the  "  Sirene,"  an  acoustic  apparatus  invented  expressly  for  this 
purpose,  by  which  the  exact  number  of  vibrations  to  every  possible  musical 
tone  is  unerringly  indicated.  It  is  asserted  by  scientists  tbat  irifhin  a  single 
Octave  from  one  to  two  hundred  distinct  degrees  of  pitch  can  be  rendered  per- 
ceptible to  the  ordinary  ear,  and  to  that  of  the  trained  violinist  a  much  larger 
number.  (See  Sedley  Taylor's  Treatise  on  the  Physical  Basis  of  Music.)  From 
these  considerations  we  may  form  an  idea  of  the  great  number  of  possible 
different  musical  sounds,  and  consequently  of  the  extent  of  the  reduction  of 
tone-material  implied  in  our  modern  musical  system. 


INTRODUCTION.  5 

characteristics,  viz:  Filch,  Duration,  aud  Force* — in  other  words, 
they  are  high,  or  low  (as  explained  above) ;  long,  or  shori  ;  loud,  or 
soft.  Moreover,  they  may  be  considered  as  separate,  or  in  combina- 
tion. Hence  the  whole  science  of  music  may  be  comprised  under 
these  four  general  heads :  1.  Melodics,  the  doctrine  of  the  com- 
position of  melody  from  tones  differing  in  pitch  and  forming  Inter- 
vals; 2.  Rhythmics,  which  treat  of  the  disposition  of  tones  in  a 
piece  of  music  according  to  their  relative  duration,  implying  the 
grouping  of  them  into  measures,  with  regular  accents,  etc. ;  3.  Dy- 
ifAirics,  whose  subject-matter  is  the  various  degrees  of  power,  i.  e., 
loudness  and  softness,  to  be  given  to  the  diflFerent  tones — and  in 
general,  what  is  called  "musical  expression;"  and  4.  Harmonics, 
the  theory  of  Harmony,  i.  e.,  of  combining  tones  so  as  to  sound 
simultaneously  and  form  chords. 

Passing  over  the  branches  of  Rhythmics  and  Dynamics,  as  not 
coming  Avithin  its  special  scope,  the  present  little  work  is  concerned 
with  the  necessary  preliminaries  to  the  study  of  Harmony  and 
Counterpomt.\     It  treats  incidentally  of  the  primary  chord-forma- 

*  Another  characteristic  of  tones  ia  Quality,  or  Color  (in  German,  Klang- 
farhe;  in  French.  Tuabre)  ;  but  tliis  forms  a  specific  division,  in  treating  of 
music.  The  great  German  authority  in  acoustics.  Professor  Helmlioltz,  baa 
demonstrated  that  the  difference  in  the  quality  of  one  and  the  same  tone  (as  to 
pitch)  on  various  instruments,  is  due  to  difference  in  the  form  of  ttie  vibrations, 
i.  e.,  in  the  outlines  of  the  sound-waves,  as  observed  by  him  with  the  aid  of  the 
microscoie. 

f  Counterpoint  may  be  called  the  art  of  combining  melodies,  in  distinction 
from  that  of  combining  tones  into  chords,  which  is  properly  called  Harmony. 
"  When  we  look  at  a  piece  of  harmonized  music  from  the  contrapuntal  \)(nnt 
of  view,  we  mostly  direct  our  attention  to  the  melodies  of  which  each  part 
should  consist. . .  .We  look  at  these  melodies  as  it  were  horizontally,  along  the 
paper,"  Inot  perpendicularly,  as  in  the  case  of  chords)  "and  the  harmonic  deri- 
vation of  the  chords  they  may  jointly  produce  is  kept  out  of  sight,"  etc.— ^> 
F.  Ouseley,  Counterpoint. 


6  INTRODUCTION. 

tions  (Triads),  thus  touching  the  fourth  branch — Harmonics,  but 
is  devoted  chiefly  to  Melodies,  and  to  this  branch,  indeed,  only  to  a 
certain  extent,  viz :  so  far  as  it  comprises  the  analysis  of  the  mate- 
rial from  vs^hich  melodies  and  harmonies  are  constructed,  {.  e.,  of 
tones,  regarded  as  differing  in  jntch,  as  modified  by  chromatic  alter- 
ation, as  variously  notated  (from  the  standpoint  of  musical  orthog- 
raphy), as  forming  the  different  Intervals,  and  as  arranged  in  the 
two  diatonic  series  or  tone-families  respectively  known  as  the 
Major  and  the  Minor  Key.  Hence  this  little  work  claims  to  be 
nothing  more  than  its  title  implies — a  Primer  of  Modern  Tonality, 
using  the  word  "tonality"  to  express  a  system  or  arrangement  of 
tones,  as  best  exhibited — in  the  case  of  modern  tonality — in  the 
Diatonic  and  the  Chromatic  Scale.*  The  usefulness  of  such  a  work, 
however,  is  deducible  from  the  fact  that  it  is  impossible — as 
almost  every  teacher  knows  by  experience  with  one  or  the  other  of 
his  pupils — to  make  solid  progress  in  the  study  of  Harmony  and 
Composition  without  a  previous  thorough  and  exhaustive  knowledge 
of  the  material  of  music,  i.  e.,  of  tones,  regarded  under  the  various 
aspects  above  indicated. 

Accordingly,  the  author  of  this  Primer  has  endeavored  to  ex- 
plain, to  the  best  of  his  ability,  step  by  step,  and  with  accuracy 

*  Thus  we  say,  speaking  in  this  sense,  that  the  "  tonality  "  of  the  plain- 
chant  or  "  Gregorian  "  melodies,  for  instance,  which  are  used  to  some  extent  in 
the  public  worship  of  the  Roman  Catholic  Church,  is  so  unlike  our  modern 
tonality  tliat  it  is  difficult,  if  not  altogether  impossible,  to  Imrmonize  most  of 
these  melodies  satisfactorily.  So,  too,  we  say  that  in  the  respective  musical 
systems  of  the  Chinesp,  the  Turks,  etc.,  there  is  a  different  "  tonality  "  from 
ours,  I.  e.,  a  different  selection  and  arrangement  of  tones,  a  different  scale,  etc., 
involving,  of  course,  melodies  of  a  peculiar  and  to  us  strange  character — just 
as,  on  the  other  hand,  our  melodies  are  strange  to  the  natives  of  barbarous,  or 
semi-civilized  countries. 


IXTRODUCTIOy.  7 

and  conciseness,  all  that  is  necessary  to  be  known  in  this  matter, 
and  to  dispel  the  confusion  of  ideas — not  to  say  the  positive  igno- 
rance— on  the  subject  of  touahty  so  prevalent  as  to  be  sometimes 
found  even  among  those  who  pass  for  "musicians."  Indeed,  in 
most  of  the  educational  works  on  music  which  have  hitherto  ap- 
peared in  English,  the  matter  of  tonaUty  is  by  no  means  exhaust- 
ively treated,  and  in  such  explanations  of  it  as  are  vouchsafed  there 
is  so  much  obscurity  of  language,  so  httle  care  to  give  exact  dejinu 
tions  of  technical  terms,  with  appropriate  exemplitications,  that  tho 
confusion  of  ideas  alluded  to  as  being  so  prevalent  is  hardly  to  b» 
wondered  at.  The  author's  experience  with  pupils  in  Harmony 
long  ago  convinced  him  of  the  necessity  of  an  elementary  treatise 
hke  the  present  work — for,  surely,  it  is  inconsistent  with  thorough- 
ness of  instruction  to  enter  upon  the  subjects  of  Chord-combina- 
tion, Suspension,  Modulation,  etc.,  with  the  student  who  cannot— 
for  example — ^give  a  scientific  definition  of  the  tonal  "  interval,"  or 
explain  the  structure  of  the  Diatonic  Scale.  It  was  thought  that 
this  little  Primer  might  be  welcome  especially  to  teachers  of  Har- 
mony and  Composition,  as  an  inexpensive  hand-book  to  be  recom- 
mended to  their  pupils  for  self-study. 

A  few  words  of  explanation  are  offered,  in  view  of  some  pecu- 
liarities of  this  work. 

Certain  expressions  will  perhaps  be  found  strange,  even  awk- 
ward— but  only,  as  the  author  believes,  because  they  are  netv.  Our 
English  musical  vocabulary  is  on  the  one  hand  miserably  defective, 
and  on  the  other  hand  encumbered  with  expressions  which  are  not 
significant,  or  do  not  clearly  define,  frequently  necessitating  awk- 
ward circumlocutions.  We  need  to  supply  the  place  of  such  ex- 
pressions by  exact,  concise  terms,  and  to  introduce  new  ivords. 


8  INTRODUCTION. 

coined  from  other  languages,  or  adapted  or  compounded  from  our 
own  tongue,  after  the  German  manner.  At  the  risk  of  incurring 
the  odium  which  attends  every  innovation,  the  author  has  endeav- 
ored to  do  something  towards  supplying  these  needs,  convinced 
that  it  is,  generally  speaking,  less  vital  to  the  interests  of  musical 
science  to  retain  established  usages  merely  hecause  estallished,  than 
to  enunciate  the  principles  of  that  science  in  exact  and  concise 
terms. 

The  system  which  acknowledges  only  two  hinds  of  Intervals — 
major  and  minor — as  normal  in  the  Diatonic  Scale,  is  that  of  the 
eminent  musical  theorist  and  teacher,  0.  F.  Weitzmann,*  and  com- 
mends itself  by  its  perfect  simplicity.  Thus,  in  comparing,  for  in- 
stance, the  several  Fourths  in  the  Diatonic  Major  Scale  of  C,  we 
find  one — -f — h — greater  by  tonal  measurement  than  every  other 
one :  how  natural,  therefore,  to  call  this  one  major,  and  the  others, 
relatively  to  this  one,  minor,  thus  following  the  same  rule  as  when 
we  compare  Seconds,  Tliirds,  Sixths,  and  Sevenths.  The  same 
reasoning  applies  to  the  case  of  what  is  called  in  this  system  the 
major  Fifth,  or  the  minor  Fifth.  Undoubtedly,  there  is  here,  aa 
in  other  similar  cases,  a  question  o^  name  chiefly,  the  thing  remain- 
ing the  same:  nevertheless,  even  in  names  there  is  sometimes  a 
choice ;  and,  as  between  two  systems  of  naming,  that  one  would 
seem  preferable  which — as  we  claim  for  Weitzmann's  system — con- 
duces to  greater  simplicity  of  doctrine  by  eliminating  such  excep- 
tions to  rules  as  are  not  founded  on  actual  necessity. 

*  Carl  Friedrich  Weitzmann  was  born  Aug.  10,  1808,  in  Berlin,  where  he 
still  resides.  His  "  Harmoniesystem  "  (Prize  Essay)  has  attracted  great  atten- 
tion throughout  the  musical  world. 


MODERN     TONALITY. 


OHAPTEB    I. 

Tones,  and  their  Notation. 


1.  The  expression  Toxe  is  used  in  this  work  to  signify  a 
musical  sound,  produced  by  the  agency  of  the  human  voice,  or  of 
a  musical  instrument,  etc.  As  every  tone  is  generated  by  a  cer- 
tain number  of  vibrations  of  an  elastic  body,  when  we  speak  of  a 
tone  we  mean  a  definite  musical  sound,  distinguished  from  other 
sounds  by  its  pitch,  i.  e.,  its  relative  acuteness  or  depth.  It  is 
exclusively  in  this  sense  that  we  shall  use  the  word  "Tone":,  to 
express  the  ideas  conveyed  by  the  popular  terms,  "  whole  tone  '* 
and  "half  tone"  (or  ''semitone"),  other  and  more  appropriate 
terms  are  employed.* 

3.  Out  of  the  multitude  of  possible  diiferent  tones  (differing 
in  pitch,  as  above  explained)  twelve  have  been  adopted  as  an  ample 
material  for  the  purposes  of  modern  musical  art,  and  their  ar- 
rangement in  a  scientific  system  constitutes  our  modern  tonality, 

*  It  is  every  way  desirable  that  the  word  "  Tone  "  should  be  used  exclusively 
in  its  proi)L'r  sense.  According  to  the  too  common  method  of  teaching,  the 
pupil  is  told,  for  instance,  to  strike  or  sing  c  (which  is  undeniably  a  tone), 
then  d  (which  is  another  tone),  and  is  informed  that  the  progression  or  inter- 
val is  a  tone !  or,  it  is  taught  that  the  tiro  tones,  e  and/,  or  h  and  e.  give  a  half- 
tone! The  English  translator  of  Dr.  Marx's  Allgcmeine  Musiklelire  is  driven 
to  the  expedient  of  ])utting  the  word  "  tone  "  in  italics  when  it  is  used  in  its 
proper  sense,  and  in  Roman  letters  when  it  means  (as  he  says)  "  distance  or 
interval"  ( ! ).     See  Note,  page  16. 


10  MO  D  E R  N     T  0  JSfA  L  IT  Y, 

as  distinguished  from  the  tonal  systems  of  antiquity.  These 
twelve  tones  are  repeated  over  and  over  again  in  a  higher  and  a 
lower  pitch,  as  shall  presently  be  explained.  Among  them,  seven 
are  classed  as  primary  tones,  and  named  after  the  first  seven  let- 
ters of  the  alphabet.  A,  B*  6',  D,  E,  F,  G.  We  shall  turn  our 
attention  first  to  these  seven  primary  tones. 

3.  Tones  are  expressed  to  the  eye  by  notes  (^,  f,  ,•  r,  5,  etc.f ) 
written  on  thf  degrees  (lines  and  spaces)  of  the  Staff,  or'system 
of  five  parallel  lines.  The  degrees  are  generally  counted  upward 
from  the  lowest  line. 

Fig.  1. 


-1^ 


-3 4, 


:r=^ 


-7- 


-g- 


4.  Each  staff-degree  takes  a  distinctive,  fixed  name,  represent- 
ing a  primary  tone,J  as  A,  C,  G,  etc.  To  this  end  certain  charac- 
ters called  Clefs  are  used,  each  one  naming  a  certain  degree,  from 
which  the  names  of  the  other  degrees  are  determined  by  following 
the  proper  order  of  the  letters.  A,  B,  C,  D,  E,  F,  G.      The  G 

*  The  Gentians  use  S"  instead  of  B,  and  B  for  what  we  call  B  pit. 

\  The  consideration  of  the  diflferent  musical  notes,  as  indicating  relative 
duration  of  tones,  belongs  to  Rhythmks. 

X  Any  staflFdegree  may,  however,  be  the  seat,  not  only  of  its  primary  tone, 
but  also  of  a  second,  third,  fourth,  or  fifth  tone,  regarded  as  a  modif  ration  of 
the  primary  tone,  consequently  written  on  the  same  degree  (see  20,  p.  19). 
Again,  one  and  the  same  tone,  as  V,  G,  etc.,  may  be  written  (with  different  names) 
on  three  different  staff-degrees.  These  things  find  their  exjjlanation  later — 
suffice  it  to  say  here,  that  there  is  great  flexibility  (so  to  speak)  in  the  staff, 
admirably  adapting  it  to  the  purposes  of  a  very  manifold  notation  of  musical 
sounds.  For,  though  a  given  tone  is,  as  such,  absolutely  speaking,  always  the 
same  (being  produced  by  a  certain  number  of  vibrations),  yet  its  iiaine  is  not 
an  absolute,  invariable  thing,  but  changes  according  to  the  laws  of  musical 
orthography — hence  its  seat  on  the  staff  is  variable  also. 


MODERy     TONALITY.  11 

(.  lef,  placed  on  the  3d  degree  {2d  line)  of  the  staff,  names  that  de- 
gree G  : 

Flff.  2. 

IT 


and  the  F  Clef  (Bass  Clef),  placed  on  the  7th  degree  (4th  line), 
names  that  degree  F : 

Fig.  3. 

-jj A 


^ 


'E' 


-D 


-B- 


The  F  Clef  is  used  in  the  notation  of  the  deejier  tones,  the  G  Clef 
in  that  of  the  liiglier  or  more  acute. 

5.  The  key-board  of  the  piano-forte  exhibits  to  us  the  entire 
range  or  compass,  from  the  highest  to  the  lowest,  of  the  tones 
most  used  in  music. 

6.  If  now  we  start  from  any  luldte  key,  and  strike  the  succes-* 
sive  white  keys  in  an  uninterrupted  ascending  order  till  we  reach 
the  eighth  above,  we  shall  find  tliis  eighth  tone  to  be  a  repetition 
of  the  first,  but  in  a  higher  pitch.*  Or,  starting  from  the  same  first 
tone  and  descending,  we  shall  find  the  tone  given  by  the  eighth 
white  key  below  to  be  a  repetition  of  the  first  in  a  lovoer  pitch.  Con- 
sequently, if  that  first  tone  is  C,  for  instance,  its  repetitions  above 
and  below  will  also  have  the  same  name,  C.     Each  repetition 

*  We  say,  a  higher  pitch  ;  for  if  the  pitch  of  this  eighth  tone  were  not 
merely  higher,  but  also  absolutely  different,  as  is,  in  respect  of  the  starting-tone, 
that  of  each  of  tlie  six  other  tones,  the  (Mi^-lith  tone  could  not  be  a  reiwtition  of 
the  first.  But,  in  fact,  when  both  the  latter  are  sound<!d  together,  the  ear  doea 
not  distinguish  absolutely  different  tones,  as  it  always  does  when  any  othef 
tone  than  the  eighth  (or  one  of  its  repetitions)  is  sounded  with  the  starting- 
tone. 


12 


MODERN     TONALITY. 


(eighth  tone)  is  called,  in  respect  of  the  starting-toue,  its  Octave, 
an  expression  which  is  also  nsed  in  a  broader  sense,  signifying  the 
whole  succession  of  tones  from  any  starting-point  to  its  repetition. 
The  following  figure  exhibits  the  greater  part  of  the  entire  com- 
pass of  the  primary  tones  used  in  music : 

Fig.  4. 


Last  half  of  the  Connter-Oetave. 

^= 

Great  Octave. 

Small  Octave. 

1= 

C    D    E   F     GAB 

^r : 

C    D    E    F    G    A    B 

1 

c     (1     e    f     g     a     b 

1 m   *   *~* — r 

—   — •   -=-   -•" 

: ^    ,    »    *- 

c     d     e     f     g     a     b 

r^-m.-^'^CDEFGAB 

Ci   B^  E^  Fi    Gi  Ai  Bi 


Thrice-maiked  (lined)  Oct. 


Once-raarked  (lined)  Octave.  Twice-marked  (line.')  Oct. 


Sva. 


i.  ^tetc. 


^^^m^=^==^^=E==Ei 


c   d    e  f   g    a    b      c    d     e    f    g     a     b     cdefgabcde 
c'd'-e'.  etc.  c"  d"  e",e:tc.  c'"  d'"  e'",  etc.  c""d""e"" 

Cj  di  fij,  etc.  c-2  d-i  e-,,  etc.  C3  d^   e^,  etc.  c^  d^  e^ 


m 


Explanation. — 1.  The  highest  and  lotve-4  tones  cannot  be 
written  on  the  staff  itself — we  have  to  use  added  lines  I'formerly 
called  "leger  lines  ")  above  and  below  it.  The  1st  added  line  below 
the  upper  staff  indicates  the  very  same  tone  as  that  above  the  lower 
staff;  viz.:  c{c',  c.^).*    2.  For  convenience  of  distinguishing  the 


*  Besides  the  F  Clef  and  the  G  Clef  there  are  three  other  Clefs,  used 
chiefly  in  Orchestral  scoring,  viz.,  the  Soprano,  fhe  Alto,&ndL  the  Tenor. 
All  have  the  same  form,  and  all  are  C  Clefs,  i.  e.,  each  one  fixes  the 
once-marked  C  (c  or  c),  above  referred  to,  on  a  different  line  of  the 
staff,  for   the  -greater   convenience  of   notation.      Thus,  music  for   Soprano 


instruments    or    voices    may,    with    the    Soprano    Clef 


»= 


{c    on    the 


MODERK     TONALITY.  13 

several  repetitions  of  one  and  the  same  tone,  the  tones  are  gi'ouped 
into  Octaves  aising  the  term  in  its  broader  sense),  starting  from 
Counter-C  (the  lowest  C  on  the  modern  piano-forte\  each  Octave 
having  a  distinctive  name,  as  in  the  above  figure.  Thus,  the  expres- 
sions, "Counter- 6'"  ;^\vritten  C,  or  C^),  "  G  of  the  Great  Octave" 
{G),  "F  of  the  Small  Octave"  (/),  "Once-marked  or  Once-lined 

D"  {d,  d',  d^),  "Twice-marked  lined)  ^"  (e,  e",  e^),  etc.,  serve  to 
indicate  these  several  tones  with  the  accuracy  of  notation. 

QUESTIONS. 

1.  What  is  the  exclusive  signification  of  the  word  "  tone,"  as  used  in  this 
work?. . .  .2.  How  many  actually  different  tones  have  we  in  modern  music? 
The  arrangement  of  these  tones  in  a  scientific  system  constitutes  what?  How 
many  of  the  tones  are  classed  as  primary?  How  are  these  latter  named  ?. . . . 
3.  How  are  tones  expressed  to  tiie  eye?. . .  .4.  Has  eacli  staff-decree  a  fixed 
name  ?  (Does  a  staff-degree  admit  the  notation  on  it  of  more  than  one  tone  ? 
See  Note.)     What  is  the  character  used  to  give  a  fixed  name  to  a  certain 

degree  of  tlie  staff?     Name  the  two  of  these  characters  most  in  use O.  In 

an  uninterrupted  succession  of  the  tones  sounded  by  the  lo/iite  keys  of  the 
piano,  ascending  or  descending  from  any  tone  as  a  starting-point,  which  tone 
is  always  a  repetition  of  the  starting-point  ?  What  is  such  a  repetition-tone 
called,  in  respect  of  the  starting  tone  ?  What  does  the  expression  "  Octave  " 
signify,  in  a  broader  sense  ?     Can  we  indicate  a  tone  out  of  the  entire  tone- 


1st  line),  be  nutated  almost  entirely  on  the  Staff,  i.  e.,  without  added  lines; 
and  the  same  thing  may  be  said  of  music  for  Alto  instruments  or  voices,  with 


the  AUo  Clef  -'^ (c'  on  the  3d  line),  and  of  music  for  Tenor  instruments 


or  voices,  with  the  Tenor  Clef  V^ (c'  on  the  4th  line).    In  the  notation  of 

vocal  music,  the  common  practice  at  present  is  to  use  only  the  clefs  of  G  and  F. 
It  may  be  ol)served  here,  that  many  Tenor  singers,  seeing  their  part  written 
on  the  staff  with  the  G  clef,  like  the  Soprano  part,  erroneously  suj)i)ose  that 
they  sing  the  notes  as  written,  whereas  they  really  sing  them  an  Octave  lower. 
Thus,  for  instance,  twice-marked  c,  wliile  sung  as  written  by  a  Soprano  voice, 
represents  (/iLce-marked  c  for  a  Tenor  voice. 


14  M  0  D  E  R  X     T  0  NA  L  I  T  T. 

compass  (Fig.  4),  a&  to  its  exact  pitch,  in  any  other  way  than  by  notation  ? 
Name  the  diffeient  Octaves  into  which  the  tones  are  grouped,  beginning  with 
the  lowest.  What  liind  of  letters  indicate  the  tones  of  the  Counter-Octave  ? 
Those  of  the  Great  Octave  ?  Those  of  the  Small  Octave  ?  Those  of  the  once- 
toarked  Octave  ?  etc.,  etc.  (What  other  clefs  are  there  besides  those  of  O  and  F? 
What  letter  does  each  indicate  ?  On  which  line  of  the  staff  does  the  Soprano 
clef  fix  this  letter  ?  On  which  the  Alto  ?  On  which  the  Tenor  ?  Which  clefs 
are  usually  employed  in  the  notation  of  vocal  music  ?  What  mistake  is  often 
inade  in  regard  to  the  actual  pitch  of  the  Tenor  part,  notated  on  the  staff  with 
the  (?clef?    See  Note.) 


CHAPTER    II. 

Scale  and  Key.     The  Diatonic  Scale  in  General. 

7.  What  we  have  hitherto  called  the  Octave,  in  its  broader 
sense,  is  more  usually  called  the  Scale  (Gernran,  "  Tonleiter,"  i.  e. 
"  Tone-ladder "},  which  may  be  defined  as  the  uninterrupted  suc- 
cession or  series  of  tones  from  any  starting-point  to  its  rej)etition 
or  Octave.  The  series  may  be  composed  of  seven  different  tones 
only,  or  may  embrace  all  of  the  twelve  tones  Avhich,  as  we  shall 
presently  see,  are  contained  within  the  compass  of  an  Octave — 
hence  the  scale  is  either  Diatonic  or  Chromatic,  the  latter  of  course 
including  the  former.  Every  scale  is  named  after  its  starting- 
point,  and  the  Octave  is  usually  added  to  complete  the  series.  The 
following  are  examples  of  Diatonic  Scales : 

Fi(f.  5.    Diatonic  Scale  of  A. 


A     B      c,  etc. 
I       II     III,  etc. 


m 


Fig.  6.    Diatonic  Sc.\le  of  G. 


c       d       e,  etc. 
I      II    III,  etc. 


MODERN     TO  NA  L  I  T  Y.  15 

Eemark. — The  stalf-degrees  occupied  by  the  tones  of  a  Dia- 
tonic Scale  are  numbered  upward  from  the  starting-point,  Eoman 
numerals  being  usually  employed.  Thus,  in  Fig.  5,  A  would  be 
marked  I,  B  II,  c  III,  and  so  on  ;  and  in  Fig.  G,  c  would  be  I,  d  II, 
and  so  on. 

8.  The  expression  Key,  in  music,  will  be  more  fully  explained 
later — it  is  enough  to  say  here  that  it  indicates  a  family  of  tones 
(so  to  speak)  standing  in  a  fixed,  subordinate  relationship  to  a 
predominant,  fundamental  tone,  called  the  Jcey-tone  (.less  properly 
*'  key-note  ").  A  piece  of  music  is  said  to  be,  for  instance,  in  the 
key  of  C,  or  of  A,  according  as  C,  or  A,  is  the  key-tone. 

9.  Every  key  is  represented,  as  to  its  essential  tone-material — 
both  melodic  and  harmonic — by  its  Diatonic  Scale  ;  thus,  the  es- 
sential tones  of  the  key  of  C,  for  instance,  are  C,  D,  E,  F,  G,  A,  B. 
But  a  piece  in  a  given  key  may  contain,  besides  the  seven  essential 
tones,  some  if  not  all  of  the  other  tones  (derivatives)  included 
"within  the  Octave.  In  this  case  we  have  the  chromatic-diatonic 
genus  of  composition,  the  warm,  expressive  style  of  modern  times. 
If,  however,  there  is  a  rigid,  exclusive  adherence  to  the  essential 
tones  of  a  key,  we  have  the  pure  diatonic  genus,  which  is  charac- 
terized by  a  certain  coldness,  stiffness,  and  quaint  simplicit}'-,  ren- 
dering it  the  style  best  adapted  for  the  composition  of  music  in- 
tended to  awaken  thoughts  of  ancient  times. 

10.  The  importance  of  the  Diatonic  Scale  renders  it  necessary 
that  we  should  examine  more  closely  into  its  structure,  in  doing 
which  we  shall  call  in  the  aid  of  the  key-board  of  the  piano-forte 
or  organ. 

11.  In  playing  a  Diatonic  Scale,  for  instance  that  of  C  (Fig.  6), 
on  the  key-board,  we  notice  an  intermediate  (black)  Icey  between  c 
and  d,  d  and  e,  f  and  g,  g  and  a,  and  a  and  i,  but  none  between  e 
and/,  and  h  and  c.  Now  the  key-board  of  the  piano  is  divided  into 
half-steps,  commonly  but  improperly  called  "  half-tones,"  or 
"semitones."  A  Half -step  is  a  progression  from  any  key  to  the 
next  above  or  leloic,  irrespectively  of  color.    From  c,  for  instance,  to 


16  MODERN     TONALITY. 

the  key  next  above  (black)  is  a  half-step,  and  from  this  black  key 
to  the  key  next  above  (white),  which  is  d,  is  another  half-step. 
Again,  from  c  to  the  key  next  below,  which  is  B,  is  a  half-step ; 
or,  from  d  to  the  black  key  next  below  is  a  half-step.  A  progres- 
sion from  one  cone  to  another,  as  from  c  to  d,  involving  two  half- 
steps,  \%  called — not  a  "Tone"  or  "whole  Tone,"  but — a  Step* 
(whole  step) ;  in  other  words,  whenever  I  pass  from  any  key  to 
the  next  hut  one  (above  or  below),  i.  e.,  skipping  one  intermediate 
hey,  white  or  black,  I  make  a  musical  or  tonal  Step.  Again':  we 
find  an  intermediate  key  between  d  and  e,  therefore  from  d  di- 
rectly to  e  is  a  Step  ;  and  similarly,  from  /  to  cj,  g  to  a,  a  to  h — all 
Steps.  But  as  there  is  no  intermediate  key  between  e  and/,  and  h 
and  c,  it  follows  that  from  e  to  /,  and  from  h  to  c,  is  a  half-step. 
Hence,  it  appears  that  the  progressions  or  changes  of  tone  in  the 
Diatonic  Scale  are  by  steps  and  half-steps,  there  being  five  of  the 
former  and  two  of  the  latter. 

13.  We  may  now  more  fully  define  the  Diatonic  Scale  i)i  gen- 
eral \  as  a  series  of  seven  different  tones  (with  the  Octave  of  the 
first  added)  following  each  other  in  such  order  that  five  of  the 
progressions  are  steps,  and  two  are  half-steps,  the  latter  being  in^ 
termingled  with  the  former. 

Eemark. — Inasmuch  as  in  the  Diatonic  Scale  each  tone  has 


*  The  greater  appropriateness  of  tliis  expression  is  seen  from  tbis,  that  the 
very  word  "  step"  at  once  suggests  the  i  !ea  of  a  progression  from  one  sound  or 
tone  to  another,  whereas  the  word  "  tone"  suggests  notliing  of  the  kind,  but 
only  one  sound,  without  reference  to  any  otlier.  The  substitution  of  the  word 
"half-step"  for  "semitone"  is  justified  by  the  same  argument,  and  even  more 
clearly,  inasmuch  as  the  average  intellect  more  readily  conceives  of  a  step 
being  halved  than  of  a  sound  undergoing  that  process.  The  sooner  we  lay 
aside  prejudice  in  favor  of  bad  usages,  how  venerable  soever,  and  employ 
terms  which  suggest  their  own  meaning,  the  better  will  it  be  for  the  future  of 
popular  musical  instruction. 

f  When  we  come  to  speak,  further  on,  of  particular  Diatonic  Scales,  a 
particular  definition  will  be  necessary  for  each. 


MODERN     TO  NA  LIT  T.  17 

its  own  peculiar  name  and  pJace  on  fJte  staff,  special  attention-  is 
here  called  to  the  significance  of  tlie  word  "diatonic."  For  in 
every  diatonic  progression  from  one  tone  to  another,  the  second 
tone  has  a  different  name  and  staff-degree  from  the  first.  The  im- 
portance of  this  will  be  apparent  further  on. 

13.  Taking  each  of  the  primary  tones  as  a  starting-point,  we 
may  construct  the  following  seven  diatonic  scales,  differing  in 
structure  in  the  relative  positions  of  the  two  half-steps.  Thus,  in 
No.  1  the  first  half-step  occurs  between  degrees  II  and  III,  and  the 
second  between  V  and  VI ;  in  No.  3  the  first  half-step  occurs  be- 
tween I  and  II,  and  the  second  between  IV  and  V ;  and  so  on  of 
the  rest.     (The  half-step  is  indicated  by  this  mark  - — - .) 

Fi(f.  7. — Diatonic  Scales.* 


I 

II 

III 

IV 

V 

VI 

VII    VIII 

No.  1. 

A 

if 

c 

d 

e 

^/ 

9         » 

No.  3. 

B^ 

c 

d 

e 

'^f 

9 

a         b 

No.  3. 

c 

d 

e 

""/ 

9 

a 

h       ^G 

No.  4. 

d 

e 

"/ 

9 

a 

h^ 

c         d 

No.  .5. 

e 

^f 

9 

a 

iT 

c 

d        ~e 

No.  6. 

f 

9 

(I 

h^ 

c 

d 

j^~y 

No.  7. 

9 

a 

h^ 

c 

d 

e 

""/      's 

14.  Of  the  foregoing  Diatonic  Scales  that  of  A  (No.  1)  and 
that  of  C  (No.  3)  are  used  in  modern  music  as  model  or  normal 
scales,  almost  to  the  exclusion  of  the  others,  which  are  obsolete 
except  in  some  kinds  of  ecclesiastical  song. 


*  Tlifsn  are  the  anfit^nt  so-rallod  ErrlemaKtlral  Muden  or  Soalos,  sii^iposed  to 
have;  been  dcrivcfl  from  the  (Jre(  k  iiiiisical  system,  and  forming  tlie  basis  of 
the  church-song  introduced  by  Pope  Gregory  the  Great  (in  the  6th  century), 
and  called  after  his  name. 


18  MODERN     TONALITY. 


QUESTIONS. 

7.  What  is  tlie  Scale?     When  is  the  scale  called  " diatonic  " 7     When 
"  chromatic  V  "     From  which  one  of  its  tones  is  every  scale  named  ?    How  are 

the  staff-degrees  in  the  scale  numbered?  (Remark.) S.  What  is  meant  by 

"  Key,"  in  music'?. . .  .9.  How  is  a  key  represented  as  to  its  essential  tone- 
material?  Give  an  example.  When  is  a  composition  in  the  •'chromatic- 
diatonic"  genus  or  style'?     When  in  the  pure  "  diatonic "  ?  and  what  is  the 

character  of  each  style  ? 11.  What  is  a  "  Half-step  "  ?     What  is  a  "  Step  "  ? 

Give    examples    of  each.     (Why  should    we   say  "  Step,"   and   '•  Half-step," 

rather  than  "  Tone,"  and  "  Semitone  "  ?    See  Note.) 12.  Give  now  a  full 

definition  of  the  Diatonic  Scale  in  general.  What  important  peculiarity  of 
every  diatonic  progression  should  be  specially  noted  ?  (Remark.). . .  .13.  How 
many  different  diatonic  scales  may  be  formed  ?  What  makes  one  differ  from 
another  ?.  . .  .  14.  How  many  of  the  seven  diatonic  scales  are  used  as  models 
in  modern  music  1    Name  them. 


OHAPTEE    III. 

Modification  or  Chromatic  Alteration  of  Tones. 

15.  Having  thus  far  considered  cliiefly  the  seven  tones 
which  are  classed  as  primary,  we  must  now  turn  our  attention  to 
our  tone-material  in  its  entirety,  including  certain  modifications 
of  tlie  primary  tones. 

16.  We  have  already  seen  that  besides  the  tones  A,  B,  C,  D, 
E,  F,  G,  we  have  intermediate  tones;  and  if  we  start  from  any 
key  on  tlie  piano  and  strike  all  the  following  ones,  white  and 
black,  in  an  uninterrupted  succession  up  to  the  Octave  of  the 
starting-point,  exclusively,  w^e  shall  find  the  whole  number  of  dif- 
ferent tones  to  be  twelve. 

17.  The  intermediate  tones  may  be  regarded  as  derivatives  of 
the  primary  tones  in  so  far  as  they  take  the  names  of  the  latter. 


MODERX     TOyALITY. 


19 


with  certain  additions  implying  tliat  the  primary  tones  are  in 
some  way  modified  or  altered,  viz.,  in  pitch. 

18.  A  tone  is  modilied  as  to  pitch  by  raising  or  by  depression. 
The  raising  of  a  tone  is  indicated  by  placing  before  its  representa- 
tive note  the  Sharp  ( ^ ),  wliich  raises  by  a  half-stejj,  or  the 
Double-sharp  (  x  ),  which  raises  by  two  lialf-steps.  The  de- 
pression  of  a  tone  is  indicated  by  prefixing  to  its  corresponding 
note  the  Flat  ( \, ),  which  lowers  by  a  half-step,  or  the  Double- 
flat  ( [^l; ),  which  lowers  by  tivo  half-steps.  The  x  indicates 
double-raising,  being  nsed  only  with  a  note  already  sharped  ;  and 
the  ]^\f  indicates  douUe-depression,  being  used  only  with  a  note 
already  flatted. 

19.  A  fifth  character,  the  Cancel  (t;]),  annuls  a  previous  |^, 
U ,  X  .  or  j^l; ,  and  the  staff-degree  affected  by  it  now  represents  its 
primary  or  natural  tone  (whence  this  sign  is  sometimes  called  the 
*' Natural").  The  Cancel  therefore  indicates  the  raising  or  the 
loivering  of  a  tone,  according  to  circumstances — the  former,  when 
neutralizing  the  [>  or  \^\^\  the  latter,  when  neutralizing  the  ^ 
or  X. 

20.  These  five  characters,  the  ^,  x,  [?,  [?[?,  and  ^,  are  called 
Accidentals,  or  better.  Chromatic  Signs,  and  they  effect  what  ia 
called  in  general  chromatic  alteration,  which,  again,  gives  rise  to 
the  chromatic  nomenclature,  or  naming  of  the  altered  tones. 
Thus,  when  the  \,  x,  I;,  or  [;[;  is  placed  on  any  staff-degree,  the 
degree  retains  its  primary  name,  but  takes  an  addition,  or  qual- 
■ification,  being  called,  for  instance,  C  sharp,  C  double-sharp,  O 
flat,  or  C  douhle-flat,  as  the  case  may  be.  In  this  way  eacli  staff- 
degree  may  be  the  seat  of  four  additional  tones,  as  in  the  following 
examples : 


Fig.  8. 


—1 ^^ tm S9 \m ^^» i n — ~~ 

4,^ Sf —    ^  "  :-!= —■ «,-_::^»__^ ^ bb* 

20  MODERN     TO  NA  LI  TY 


QUESTIONS. 

16.  How  many  diflbrent  tones,  counting  also  the  intermediate  tones,  are 
contained  within  an  Octave?. ..  .17.  In  what  sense  are  the  intermediate 
tones  derUatiees  of  the  primary  tones?.  ...  18.  How  is  a  tone  modified  as  to 
pitch?  What  does  the  Sharp  indicate?  The  Double-sharp?  The  Flat? 
The  Double-flat  ?  When  is  the  Double-sharp  used  ?  When  the  Double- 
flat  ?....!  9.  What  is  the  effect  of  the  Cancel?  Does  the  Cancel  raise,  or 
lower  ?.  .  .  .20.  What  is  the  general  expression  for  the  effect  of  the  chromatic 
signs  ?    How  does  the  chromatic  alteration  of  tones  affect  their  names  * 


CHAPTER     lY. 

The  Chromatic  Scale. 


21.  A  series  of  twelve  different  tones,  ascending  or  descending 
by  consecutive  half-steps  (the  Octave  of  the  starting-tone  being 
generally  added,  to  complete  the  series),  is  called  a  Chromatic — 
more  strictly,  CHROMATic-DiATO]!fic*— Scale.  This  Scale  is  in 
fact  a  Diatonic  Scale  with  the  intermediate  or  derivative  tones 
added  to  it,  the  latter  being  an  effective  means  for  embellishing 
and  enriching  both  melody  and  harmony,  thus  obviating  the  com- 
parative stiffness  and  monotony  necessarily  attendant  upon  ex- 
clusively diatonic  progression. 

23.  The  Chromatic  Scale  is  usually  Avritten  in  two  ways ;  viz. : 
as  an  ascending  series  with  the  chromatic  signs  of  raising,  and  as  a 
descending  series  with  those  of  depressio7i,  as  in  the  following  ex- 
amples, in  which  the  primary  tones  are  represented  by  white 
notes,  to  distinguish  them  from  the  derivative  tones,  written  in 
black  notes : 


*  See  Remark  to  24. 


M  0  D  E  R  X     TO  XA  L  I  T  Y.  21 

Fuj.  .'>.— Chromatic  Scale  op  C. 


Ascending. 


H^- Z E5 ^ ^ ^ ^ ^*"^ 

-«^ 

=^= 

1^ 

-^ 

ps^ ■f::^      t* 2 3^ :: 

c        c$      d        dt     e        f       ft     g        gt 

Descending  (to  be  read  backward  from  the  right). 

a 

at 

hm 

J) 

c' 

db     d        eb      e        f       g^     ff        ab      a        Vo 
Fig.  10. — Chromatic  Scale  of  A. 


Ascending. 


1"^'                                                                     ^              T-            '^ i»          --^ 

-^s- 

zzi-n 

=2= 

-Jt«= 

—& T, 

A      A%     B       c        ct      d       dt     e 

Descending  (to  be  read  backward  from  the  right). 

ti ;=: bm        ^ 

f 

ft 

9 

Qt 
1^ 

a 

_^ ^ ^ c- P* 



^ 

A       B*)    B        c        db     d        So      e        f       ^      g       (A>      a 

23.  The  theory  of  the  chromatic  nomenclature,  as  above  il- 
lustrated, may  he  thus  summed  up :  each  iutermediate  tone  is 
regarded  as  a  raising,  in  the  ascending  series,  and  as  a  lowering, 
in  the  descending,  of  the  primary  tone  sounded  immediately  before, 
consequently  takes  tlie  name,  with  the  suitable  addition  to  it,  of 
the  tone  of  which  it  is  a  modification.* 

Two-fold  Distinction  of  Half-Steps,  and  of  Steps. 

24.  It  is  now  proper  to  state  that  there  are  t7i'0  kinds  of  Ilalf- 
steps,  viz. :  Diatonic  and  Chromatic — a  distinction  which  is  im- 
portant, yet  involves  no  difTiculty.  Illustrations  are  found  in  the 
above  figures  9  and  10.  In  the  diatonic  half-ste])  the  second  tone 
takes  another  name,  and  is  written  on  the  stajf-degree  imme- 
diately uhove  or  below  that  occupied  by  the  first  tone  {e.  g.,  c^d, 

*  See  Appendix  I,  p.  83. 


22  MODERN     TO  NA  LIT  Y. 

d^e,  e-f,  etc.) ;  in  the  cliromatic  half-step  the  second  tone  stands 
on  the  same  staff-degree  as  the  first,  and  has  consequently  the 
same  name,  with  a  qualification  added  to  it  {e.  g.,  c-c^,  «-f^^,  etc.). 
One  and  the  same  tonal  progression  (half-step)  is  capable  of  a  two- 
fold musical  orthography;  thus,  the  progression  from  any  (',  for 
instance,  to  the  tone  immediately  above  may  be  written  C-6  |f 
(chromatic  half-step),  or  C-D^  (diatonic  half-step),  according  to 
circumstances.*  It  should  be  well  understood  that  it  is  not  the 
mere  use  of  the  cliromatic  signs  which  makes  the  cliromatic  half- 
step,  since  from  F^  to  G,  or  from  B^  to  A,  is  as  much  a  diatonic 
half-step  as  from  E  to  F,  or  from  c  to  B,  because  in  each  case  the 
name  and  staff'-degree  are  clianged — a  peculiarity  of  every  diatonic 
progression,  to  which  we  have  already  called  attention  (see  Re- 
marh  to  13). 

Remark. — From  the  two-fold  distinction  of  half-steps  as  dia- 
tonic or  chromatic,  it  will  be  seen  why  the  scale  generally  called 
"  chromatic  "  is  more  properly  called  "  chromatic-diatonic."  The 
diatonic  scale  progresses  by  diatonic  degrees  exclusively;  the  chro- 
matic scale,  however,  does  not  progress  by  chromatic  half-steps 
exclusively,  but,  as  the  examples  show  (Figs.  9  and  10),  by 
diatonic  and  cliromatic  half-steps  intermingled.  Hence  the 
greater  fitness  of  the  name  "chromatic-diatonic,"  or  "diatonic- 
chromatic." 

25.  The  Step  may  also  he  written  diatonically  or  chromati- 
cally. Thus,  e.  g.,  the  progression  g-a  (diatonic  step)  may  also  be 
written  g-gx  (chromatic  step).  However,  the  step  mostly  used  is 
the  diatonic,  and  it  is  in  this  sense  only  that  we  shall  speak  of  it 
henceforth  :  it  is  composed  of  a  diatonic  plus  a  chromatic  half- 
step,  or  vice-versa,  as  in  the  following  examples,  in  which  the  sign 
I —  distinguishes  the  chromatic  half-step  from  the  diatonic  ^^. 

*  This  is  explained  by  the  law  of  correct  musical  notation  (musical  orthog- 
lapliy),  a  subject  of  which  we  have  briefly  treated  in  Appendix  II,  p.  8& 


M  0  D  E  R  X     TO  NA  LITY.  33 

Fi(j.  11. 


i 


i=fea 


-^     ^        Iz 


=}&=: 


=:&2= 


Exercises  on  Fig.  9. 

1.  Go  through  all  the  half-steps,  first  ascending,  then  descend- 
ing, and  distinguish  the  kind  of  each,  i.  e.,  whether  diatonic  or 
chromatic. 

2.  Point  out  all  the  diatonic  steps,  first  ascending,  then  de. 
scending  {e.  g.,  not  only  the  steps  c-d,  d-e,  etc.,  hut  also  fj|-^j^, 
and  all  the  others  involving  the  use  of  the  chromatic  signs). 

QUESTIONS. 

21.  What  is  tlie  Chromatic  Scale?  Does  it  include  the  Diatonic  Scale? 
...  .22.  How  is  the  Chromatic  Scale  usually  written  ?. . .  .23.  Give  a  sum, 
mary  of  the  theory  of  the  Chromatic  nomenclature  illustrated  in  Figs.  9 
and  10.  . .  .24.  How  many  kinds  of  Half-step  are  there?  What  is  the  dis- 
tinction between  them  ?  Does  the  use  of  chromatic  signs  in  a  half-step  neces« 
sarily  make  it  a  chromatic  half-step  ?  Give  examples  of  diatonic  half-steps 
written  with  chromatic  signs.  The  Diatonic  Scale  progresses  by  diatonic 
degrees  exclusively  ;  the  Ghromutic,  not  by  chromatic  half-steps  exclusively, 
but  by  diatonic  and  chromatic,  intermingled ;  what  name,  therefore,  is  most 
appropriate  for  the  latter  scale?  (Remark.).  ..  .25.  Give  an  example  of  a 
chromatic  Step.    Which  kind  of  Step  is  mostly  used  ?    How  is  it  formed  ? 


24  MODERN     TONALITY. 


CHAPTER    Y. 

Intervals  in  the  Diatonic  Scale. 

26.  We  have  hitherto  considered  but  two  different  tonal  pro- 
gressions— the  Half-step  and  the  Step;  we  must  now  turn  our  atten- 
tion to  greater  ones.  Every  progression  from  one  tone  to  another, 
involving  a  change  of  name  and  of  staff-degree,  forms  what  is  called 
an  Interval;  in  other  words  the  Interval  is  the  result  of  such  a 
progression  from  one  tone  to  another  as  involves  a  change  of 
name  and  of  staflf-degree.*  The  term  "  Interval "  tlierefore  sup- 
poses two  different  tones,\  and  has  reference  to.their  degree-rela- 
tionsJdp,  or  relative  positions  on  the  staff.  In  writing,  e.  g.,  the 
two  tones  c,  d,  I  place  d  on  the  degree  next  above  c,  and  the 
c-degree  being  counted  as  1,  the  next  above  will  be  2,  and  I  say 
that  d  is  a  "Second"  to  c  {L  e.,  above  c,  as  we  generally  count 
upward  from  the  lower  tone.  The  word  "above"  is  generally 
understood,  but  in  counting  doimiward  from  the  upper  tone  the 
word  "below"  or  "lower"  must  be  expressed).  Again,  I  write 
c,  g,  aud  since  the  ^-degree  is  the  fifth  above  c,  I  say  that  g  is  the 

*  Hence  the  Octiwe  cannot  be  said,  strictly  speakinor,  to  form  with  its  lower 
tone  an  Interval.  "  The  Octave,"  says  the  late  Dr.  Moritz  Hauptmann,  in  his 
Harmonik,  "  irajn-esses  us  as  a  repetition  of  the  same  tone  in  a  higher  pitch. 
It  has  no  special  significance  for  harmony,  and  is,  in  this  sense,  not  a  liar 
monic  interval,  as  is  the  Fifth  or  the  Third,"  etc.  Similarly,  what  is  some- 
times classed  as  an  altered  Interval,  viz.,  the  "  augmented  Prime,"  as  c-cH,  etc., 
does  not  appear  in  this  work  among  the  Intervals.  For  as  c-c,  is  merely  a 
Unison,  r-c$  is  simply  a  chromatic  half-step,  or  an  altered  Unison. 

+  It  is  as  confusing  and  incorrect  as  it  is  common,  to  apply  tho  word 
"  Interval"  to  one  of  two  tones,  as,  e.  g.,  in  the  case  of  the  Second,  c-d,  to  call  d 
the  "  upper  interval."    Why  not  upper  tone  (of  the  interval)  ? 


MODERN     TOyALlTY.  25 

"Fifth  "  to  c,  or,  counting  downward,  tliat  c  is  the  '"'  Fifth  below" 
g  ('•  lower  Fifth  "  to  g). 

Fig.  12. 


^—^w--^'— 

, 4 — 5« 

^"-^  ^  *  0      H 

27.  We  distinguish  in  the  Interval  two  elements — Denomina- 
Tiox  i;nd  KiXD.  The  scale  affords,  within  a  compass  of  thirteen 
staff-degrees*  (involving,  of  course,  some  added  lines),  six  different 
Dexomixatiojts  of  Intervals,  viz.  :  the  Secoj^d,  the  Third  (or 
Tierce),  the  Fourth  (or  Quart;,  the  Fifth  lor  Quint),  the  Sixth 
(or  Sext),  and  the  Seventh  (or  Sepfiina).\  Each  of  these  terms 
expresses  the  relative  positions  of  two  different  tones  on  the  Staff, 
fls  explained  above. 

2S,  Each  Dexomination  of  intervals  in  the  Diatonic  Scale 
is  tiL'ofohl  in  Kind,  viz. :  either  minor  (lesser),  or  major  (greater). 
These  are  the  normal  intervals,  as  found  in  the  Diatonic  Scale  j 
or,  to  put  it  differently,  this  Scale  contains  only  two  Kinds  of 
intervals,  viz.,  minor  and  major,  of  the  several  denominations. 
A  major  interval  is  greater  by  a  cliromatic  half-step  (^24)  than  the 
minor  interval  of  the  same  denomination.     For  instance,  given 


*  If  we  did  not  go  beyond  tlie  Octave,  say  in  the  Diatonic  Scale  of  C, 
we  could  not  fonn  the  Seventh  to  any  tone  higher  than  D,  nor  the  Sixth  to 
any  tone  above  E,  nor,  in  short,  the  TJdrd,  Fourth,  and  other  intervals  abova 

b' 

f  It  would  seem  desirable  to  have  distinct  names  (as  in  German)  for  the 
Intervals,  rather  than  the  ordinals,  Third,  Fourth,  Fifth,  etc.,  and  for  this  pur 
l>ose  the  names  joriven  above  as  o|)tional,  %iz.,  Tierce,  Quart,  Quint,  Sext,  and 
Septima,  are  suggested.  The  ad()i)ti()n  of  them  may  be  urged  at  least  in  ths 
formation  of  compound  words.  For  instance,  the  expressions,  "  Tierce- r^si- 
tion,"  "  Qnint-pnsition  "  (of  the  Triad),  are  surely  less  confusing  than  "  Third- 
position,"  "Fifth-Position";  again.  "  tierce-rdatcHl,"  "quint-related,"  '•tierce- 
relationship,"  "quint-relationship,"  are  less  cumbersome  expressions  than  "  re 
lated  by  the  Third,"  etc.,  "  relationship  by  the  Fifth,"  etc. 


26 


MODERN     TONALITY. 


two  diflFerent  progressions,  say  B-f  and  c-g,  the  number  of  slaif- 
degrees  to  be  counted  will  in  each  case  be  found  the  same  (5),  ao 
that  in  each  case  we  have  a  Fifth.  So  much  for  the  denomina- 
tion of  the  two  intervals.  We  determine  the  kind  of  each  Fifth 
by  counting  the  half-stejjs  involved  in  each.  Thus,  B-f  contains 
one  half-step  less  than  c-g  ;  therefore  B-f  is  a  minor  (lesser ;,  c-g  a 
major  (greater)  Fifth.  To  make  it  clear  that  m  this,  as  in  every  sim- 
ilar case,  the  difference  between  minor  and  major  is  that  of  a  chr'o- 
matic  half-step,  the  followiug  illustrations  are  given  (Figs.  13  and 
14),  from  which  it  is  plainly  seen  that  while  the  Fifth,  B-f,  con- 
tains the  same  number  (4)  of  diatonic  half-steps  (marked  ^^ )  as 
the  Fifth,  c-g,  the  number  of  its  chromatic  half-steps  (marked  i — i ) 
is  styialler  hy  one. 

Fig.  13. — MixoR  Fifth. 


z$m •— S*^ 


Fig.  14. — Major  Fifth. 


■^-—T,=^       ^~g»= 


-m—^—^ 


This  can  be  made  still  clearer,  if  necessary,  in  the  following  way : 
we  take  this  same  minor  Fifth,  B-f,  and  make  it  major  by  adding 
a  half-step  to  it.  But  what  kind  of  half-step  ?  If  we  add  a  dia- 
tonic  half-step,  we  obtain  the  following  results: 

Fig.  15. 

Piatoiiic  Half-step  added  abnve.  Diatonic  Half-step  added  lielow. 


^E 


=J»=:*=i?^ 


*: 


=35=: 


but  in  either  case  we  have  no  longer  a  Fifth  ;  whereas,  by  adding 
a  chromatic  half-step,  as  in  Fig.  16,  the  interval  remains  a  Fifth, 
being  merely  changed  in  Mud,  i.  e.,  from  minor  to  major,  the 
proof  of  which  is  that  it  now  contains  the  same  number  of  half- 
steps  (7)  as  the  major  Fifth,  c-g  (Fig.  14). 


MODERN     TONALITY.  37 

Fig.  16. 

Chromatic  Half-step  adiloil  above.  Chromatic  Half-step  added  below. 


^=^i^^*3^=^^^=te^^i^$i^^: 


29.  We  proceed  to  examine  the  various  Intervals,  drawing 
our  illustrations  from  the  Diatonic  Scale  of  C,  and  beginning  with 

Seconds. 

30.  Any  two  different  tones,  written  on  adjacent  degrees, 
form  a  Second. 

Eemakk.— It  is  said,  "any  two  different  tones,"  for  two  tones 
might  appear  on  adjacent  degrees  without  forming  a  Second,  or 
any  progression  whatever,  as  in  Fig.  17,  which  is  an  example  of 
"  Enharmonic  change,"  to  be  explained  later. 

Fiy.  17. 


it 


■±MZ 


Jt 


'-=S»- 


31.  The  Diatonic  Scale  of  C  contains  the  following  Seconds  : 

C-D,  D-E,  E-F,  F-G,  G-A,  A-B,  B^c.  The  progressions  E-F 
and  B-c  are  diatonic  half-steps,  all  the  other  progressions  are  steiJS. 
"We  have  seen  (35)  that  the  step  is  composed  of  a  diatonic  plus  a 
chromatic  half-step  ;  hence  the  Seconds  E-F  and  B-c  are  lesser  or 
minor,  all  the  others  being  major  Seconds. 

Remark. — Every  diatonic  step  is  a  major  Second,  and  con- 
versely, every  major  Second  is  a  diatonic  step. 

33.  The  minor  Second  is  the  smallest  of  all  the  intervals, 
being  simply  the  diatonic  half-step. 

Remark. — Though  every  minor  Second  is  a.  half-step  (diatonic), 
not  every  half-step  is  a  minor  Second.  In  the  chromatic  half-step 
the  two  tones  stand  on  the  same  degree,  therefore  cannot  form  a 
Second,  as  appears  fnjm  iJO. 

33.  A  progression  by  Seconds,  minor  or  major,  is  called  grad' 


28  MODERN     TONALITY. 

ual  [i.  e.,  from  one  degree  to  the  next  above  or  below),  as  distin 
guisbed  from  one  in  Avhich  one  or  more  degrees  are  passed  ovei 
which  is  called  a  shipping  progression,  or  simply  a  skip  or  Ica;^ 
To  the  latter  class  belong  of  course  the  remaining  Intervals  to  b( 
considered. 

Thirds. 

34.  We  find  in  the  Diatonic  Scale  of  C  the  following  Thirds : 
C-E  (two  steps),  D-F  (step  and  half-step),  E-G  (half-step  and 
step),  F-A,  G-B  (in  each  two  steps),  A-c  (step  and  half-step), 
B-d  (half-step  and  step).  The  progressions  D-F,  E-G,  A-c,  and 
B-d,  being  severally  a  half-step  smaller  than  the  others,  are  minor, 
all  the  others  are  major  Thirds. 

Fourths. 

35.  The  Diatonic  Scale  of  C  gives  the  following  Fourths: 
C-F,  D-G,  E-A,  F-B,  G-c,  A-d,  B-e.  It  will  be  found  that 
one  of  these  progressions,  viz. :  F-B,  involves  three  steps,  and  every 
other  only  two  steps  and  a  half-step.  Therefore  the  Fourth  F-B 
is  major,  all  the  others  are  minor. 

Eemark. — The  minor  Fourth  is  sometimes  called  the  "  per- 
fect," and  the  major  the  "augmented"  Fourth.  As  to  the  former 
appellation,  see  Note,  p.  53 ;  as  to  the  latter,  Remark  2,  p.  46. 

Fifths. 

,*>6.  The  Diatonic  Scale  of  C  gives  the  following  Fifths :  C-0, 
D-A,  E-B,  F-c,  G-d,  A-e,  B-f.  The  last  progression,  B-f  con- 
tains two  steps  and  two  half-steps,  each  of  the  others  three  steps 
and  a  half-step.  B-f  is  therefore  the  7ninor  Fifth,  all  the  others 
are  major. 

Remark  1. — The  minor  Fifth  is  sometimes  called  the  "dimin- 
ished," and  the  major,  the  "perfect"  Fifth.  As  to  the  former  ex- 
pression, see  Remark  2,  p.  45  ;  as  to  the  latter.  Note,  p.  53. 

Eemark  2. — The  Fifth  is  a  very  important  interval,  especially 


J/  ODE  li  ^'     T  O  XA  LITY.  29 

in  harmony.     Two  tones  forming  the  7najor  Fifth  are  in   the 
closest  relationship  to  each  other  after  that  of  the  -Octave. 

Sixths. 

37.  In  the  Diatonic  Scale  of  C  are  the  following  Sixths  :  C-A, 
D-B,  E-c,  F-d,  G-e,  A-f,  B-g.  The  progressions  E-c,  A-f,  and 
B-g  are  minor  Sixths,  being  severally  formed  by  tlivee  steps  and 
two  half-steps  ;  while  each  of  the  others — major  Sixths — involves 
four  stejjs  and  one  half-step. 

Sevenths. 

38.  The  Diatonic  Scale  of  C  gives  the  following  Sevenths  : 
C-B,  D-c,  E-d,  F-e,  G-f,  A-g,  B-a.  Of  these  C-B  and  F-c  are 
major  Sevenths,  as  involving  each  a  half-step  more  than  any  one 
of  the  others,  which  are  therefore  minor  Sevenths.* 

The  "Half-step  Formula"  of  Intervals  In  the  Diatonic  Scale. 

31).  The  Half-step  Formula  is  a  simple  and  sure  method  of 
determining  an  interval  as  to  kind.  Given  two  different  tones 
written  on  the  Stafi",  the  denomination  of  the  interval  is,  as  we 
have  seen,  determined  by  the  number  of  staff-degrees  counted, 
from  the  lower  to  the  upper  tone  (or  vice-versa) ;  but  the  kind  of 
interval,  i.  e.,  whether  it  is  minor  or  major,  is  determined  by  the 
number  of  tones,  counting  all  the  intermediate  tones  from  the 
lower  to  tlie  upper  tone  inclusive. 

40.  The  latter  process  gives  a  fixed  formula  for  each  hind  of 
interval,  e.  g.,  one  for  the  miiior  as  distinguislied  from  the  major 
Third,  another  for  the  minor  as  distinguished  from  the  major 
Fourth,  and  so  on.     For  instance,  looking  at  the  Chromatic  Scale 


*  Tlie  author  of  a  new  musical  system  published  in  Geniiany  a  few  j-ears 
ago,  advocates  a  doctrine  of  Intervals  which  differs  from  that  usually  taught 
(as  represented  in  our  Primer)  in  the  distinction  of  intervals  as  eitber  "abso- 
lute" or  "  casual."     See  the  first  part  of  Appendix  III,  p.  91. 


30  MODERN     TONALITY. 

exemplified  in  Fig.  9,  p.  21,  and  comparing  the  major  Tliird  c-e 
with  the  minor  •Third  d-f,  we  find,  if  we  count  all  the  intermediate 
tones,  that  e  (the  Third  to  c)  is  the  fifth  tone  above  c,  while/  (the 
Tliird  to  d)  is  the  fourth  tone  above  d.  Every  other  major  Third 
will  be  found  to  differ  from  every  other  minor  Third  in  the  same 
way,  i.  e.,  is  a  chromatic  half-step  greater.  Hence  we  say  thab 
the  formula  of  every  minor  Third  is  4,  and  that  of  every  major 
Third,  5. 

41.  This  is  called  the  "  Half-step  Formula,"  because  we  count 
from  the  lower  to  the  upper  tone  of  an  interval  by  consecutive 
half-ste^DS  (diatonic  and  chromatic,  intermingled) :  of  course,  how- 
ever, the  actual  numher  of  half-steps  will  be  less  hy  one  than  the 
formula-number,  as  the  following  examples  clearly  show  : 

Fig.  IS. — Major  Third.    Formitla:  5. 

1st  Tone,  2d  Tone.  3d  Tone.  4tli  Tone.  5th  Tone. 


m 


Half-stepTi*   Half-step.    *"  Half-step.  **   Half-step. 


12  3  4 

Fig.  19. — Minor  Third.    Formula:  4. 

1st  Tone.  2d  Tone.  3d  Tone.  4tli  Tone. 


^  Half-step.  *•■   Half-step.      "  Half-step. 


z^^ 


4:2.  The  following  are  the  formulas  for  the  different  normal 
intervals  in  the  Diatonic  Scale,  with  one  example  of  each,  in  illus- 
tration. The  intermediate  tones  are  supposed  to  be  counted  only, 
not  sounded,  hence  they  are  i^rinted  in  small  black  notes.  A 
Eoman  numeral  is  added  every  time  a  staff-degree  is  changed,  the 
degree  on  which  the  first  tone  stands  being  always  numbered  I  in 
determining  an  interval,  whether  we  count  upward  or  downward 
from  it. 


M  0  D  E  E  X     TO  XA  LIT!' 


31 


Fig.  ;20.— Majok  Second.    Formula:  3 
13       3 


PS 


1" 


II 


I        II 


(Diatonic  Step.) 


Fiff.  ;?i.— Minor  Third.    Formula:  4. 
12      3       4 


I — '     II   III        I    m 


Fig.  22.— Major  Third.    Formula:  5, 
12      3      4       5 


I 


^^-,     tt*=^=?g^ 


II 


III 


I      III 


Fig.  23. — Minor  Fourth.    Formula:  6. 
12      3      4      5       6 


E 


iS*; 


:3«^ 


I 


II 


III     IV 


I      IV 


Fig.  ^4.— Major  Fourth.    Formula:  7. 
12       3      4       5       6       7 


Fig.  2.5.— Minor  Fifth.    Formula  :  7. 
13      3       4       5       6       7 


i 


w 


E=jJ»=r^3*z 


II 


III 


IV     V 


Fig.  20. — Major  Fifth.    Formula  :  8. 
12345678 


i 


■^ — tf* — *  — s*— 
I         II        III 


iSir== 


■       oi 


in; 


IV     V 


32 


MODERN     TONALITY. 


Fig,  ;2  7.— Minor  Sixth.     Formula  :  9, 
123456789 


=«»= 


^^- 


=i::^2=T 


I       II 


III 


IV 


V     VI 


VI 


Fi(j.  ^<9.— Major  Sixth.     Formula:  10. 
123456789      10 


f=7 


II 


Ill  IV     V 


VI 


I      VI 


Fig.  ^.9.— Minor  Seventh.    Formila:  11. 
1       28456789      10     11 


--P^ 


zj^-^^m-=tmz 


-JM=timZ 


II 


III 


IV 


V     VI 


VII 


I     VII 


Fig.  30. — Major  Seventh.    Formula  :  13. 
2      3       4       5       6      7       8      9     10     11     12 


:«*- 


II 


=«»= 


=^=j»i 


:c2z 


~g?~ 


III 


IV      V 


VI 


VII 


I      VII 


QUESTIONS. 

26.  What  is  an  Interval  ?  .How  many  tones  are  involved  in  an  interval, 
and  what  relationship  between  them  does  the  expression  indicate  ?  Give  an 
example. ..  .27.  What  two  elements  do  we  distinguish  in  the  interval? 
Name  the  various  Denominations  of  intervals  in  the  Scale.  .  .  .28.  How  many 
Kinds  of  intervals  do  we  find  in  the  Diatonic  Scale?  How  does  a  Jtuijoi^ 
interval  differ  from  the  ?Hitt(9r  interval  of  the  same  denomination  ?  Give  an 
example.  . .  .30.  How  is  the  Second  formed  ?  Can  we  have  two  tones  on 
adjacent  decrees  without  ha\inff  a  Second  ?     Give  an  example.     [Remark.) 

31.  Which  are  the  OT?!«or  Seconds  in  the  Diatonic  Scale  of  C?     What 

interval  does  the  Diatonic  ;S;^;5  constitute  ?  (Remark.).  ...32.  Which  is  the 
smallest  interval,  and  with  what  kind  of  half-step  is  it  identical  ?  Why  is  the 
chromatic  Half-step  not  identical  with  the  minor  Second?  (Remark.).... 
33.  What  is  a  gradual  progression?  What  is  a  skipping  progression  or 
leap  ?    Name  the  different  intervals  in  the  Diatonic  Scale  of  U,  beginning  with 


MOB  £  R  X     TO  iV.l  L  IT  T.  33 

34.  the  minor  Thirds;  the  major  Thirds; 35.  the  minor  Fourths; 

the  major  Fourth.     What  other  name  has  tlie  minor  Fourth  ?    What  other 

the  major  Fourth?  {Remark.) 30.  ISame  the  minor  Fifth  in  the  Diatonic 

Scale  of   C;    the  major  Fifths.      What  other  name  has  the  minor  Fifth  ? 
What  other  the  major  Fifth V  {Remark.). ..  .37.  Name  the  minor  Sixths  in 

the  Diatonic  Scale  of  C;  the  major  Sixths  ; 38.  the  minor  Sevenths  ;  the 

major  Sevenths .  . .  .31).  What  is  the  "  Half-step  Formula "'  ? -40.  Give  an 

example  of  its  use -41.  In  the  use  of  this  Formula,  will  the  number  of 

half-steps  correspond  with  the  formula-number  ? 


CHAPTEE     VI. 

The    Equal    Temperament. 


43.  We  have  thus  far  treated  of  the  intervals  in  the  Scale  on 
the  supposition  that  they  are  all  tuned,  in  keyed  instruments,  with 
mathematical  correctness.  This  assumption  is,  however,  false, 
with  one  exception,  and  the  present  chapter  is  devoted  to  the  im- 
portant subject  of  the  so-called  "  Equal  Temperament,"  or  modern 
system  of  tuning. 

44.  We  have  seen  that  we  have  12  tones,  chosen  from  a  mul- 
titude of  possible  ditferent  tones.  Now,  the  selection  of  these 
particular  12  tones  is  based  on  the  principle  of  their  inter-rela-' 
tionship.  The  nearest  tone-relationship  is  that  of  th.Q  perfect  Oc- 
tave, the  next,  that  of  the  perfect  Fifth. 

45.  Two  tones  form  a  perfect  Octave  when  their  viljrations 
give  the  mathematical  proportion  2  :  1,  i.  e.,  when  the  nppcr  tone 
vibrates  ttoice  while  the  lower  or  fundamental  tone  vibrates  once. 

46.  Two  tones  form  a  perfect  Fifth  when  their  vibrations  give 
the  proportion  3  :  2,  the  upper  tone  vibrating  three  times  while 
the  fundamental  is  vibrating  tivice. 

Kemabk. — The  perfect  Fifth  may  be  heard  in  the  so-called 


34  MOBEEX     TONALITY. 

"harmonics"  or  "over-tones"  of  a  vibrating  string,  e.  g.,  of  the 
piano-forte.  If,  first  raising  the  dampers,  we  strike  a  key — say  C 
(of  the  great  Octave),  and  Hsten  attentively,  we  shall  hear  (not  to 
mention  other  over-tones)  ascending  from  the  fundamental  C, 
first  the  perfect  Octave,  c,  then  g,  its  perfect  Fifth.  A  string  per- 
fectly tuned  to  this  c  would  give  2  vibrations  to  1  of  C ;  a  string 
perfectly  tuned  to  this  g  would  give  3  vibrations  to  2  of  c. 

47.  Now,  by  multiplying  a  tone  by  the  perfect  Octave  we  ob- 
tain a  series  of  repetitions,  i.  e.,  of  tones  of  the  same  sound  cvnd 
name,  in  a  higher  and  lower  pitch,  for  instance,  C,  C,  c,  c,  c",  c", 
etc.  Then  Ave  adopt  the  next  relationship,  and  multiply  a  tone — 
say  C— by  the  perfect  Fifth,  thereby  obtaining  the  following  sys- 
tem of  different  tones,  quint-related*  to  each  other : 

16        15        14        13         13       11        10       9        8       7 
.     .     .    fW    c\^\^    9'?^    ^9 
6       5       4       3       2       12 
4    a\,    c\,    h\>    F    C     G 

13      13      14         15        16 

4    l^   fx     cx    gx 

ExPLAXATiois".— Begin  at  C:  reading  to  the. left  we  have  the 
perfect  Fifths  below ;  to  the  right,  the  perfect  Fifths  above  C. 

48.  On  examining  the  above  series  of  tones  we  find  that  we 
have  lost  the  Ocfave-s  we  had  gained  (see  47)— neither  C,  nor  any 
other  tone  occurs  more  than  once ;  and  we  might  continue  the 
series  of  perfect  Fifths  ad  infinitum  without  ever  obtaining  repe- 
titions! Here  then  we  have  a  superabundance  of  35  different 
tones  (7  primary  tones,  each  admitting  5  modifications;  thus, 
7x5=35— which  is  as  far  as  we  can  go  without  using  the 
triple  J  or  [7),  yet  a  series  of  perfect  Octaves  cannot  be  had,  for 
hj^othetically  no  one  of  these  35  tones  is  a  repetition  of  any 
other,   but   every   one   is  absolutely   different  from  every  otLei 


"V?  <^v? 

^t^b    /■[?    4    9"? 

3       4       5 

6       7        8       9       10      11 

DAE 

B  A  4  9^  4  «# 

17         18 

19         20 

f/x      «X 

ey.     hx     .... 

*  See  the  second  Note,  p.  25. 


MODERN     TOSALITY.  35 

one,  hence  has  its  own  separate  name.     Yet,  perfect  Octaves  we 
must  have. 

49.  Xow,  modern  music  takes  this  superabundant  tone-mate- 
rial and  reduces  it  to  a  system  of  12  different  tones,  so  arranged 
that  every  13th  tone  from  any  starting-point  sounds  the  perfect 
Octave  of  that  starting-point.  Thus,  our  modern  tonal  system 
consists  of  a  series  of  perfect  Octaves,  each  Octave  embracing 
within  its  limits  12  absolutely  different  tones.  This  result  is  ef- 
fected by  means  of  a  compromise  (so  to  speak)  on  the  part  of  the 
intervals,  whereby  they  give  up  some  of  their  normal  mathemati- 
cal perfection.  The  Fifths,  for  instance  (which  are  principally 
regarded  in  tuning),  are  somewhat  flatted,*  i.  c.,  a  fundamental 
being  assumed,  the  vibrations  of  the  upper  tone  (Fifth)  are 
slightlv  reduced  in  rapidity.  We  have  seen  that  by  tuning  Fifth 
after  Fifth  perfectly  we  render  a  perfect  Octave  impossible.  For 
instance,  starting  from  Cj  and  tuning  by  perfect  Fifths,  we  shall 
not  meet  a  tone  sounding  like  that  C  iln  a  higher  pitch,  of  course) 
till  we  reach  the  12th  perfect  Fifth  above,  which  is  Zf^lj:  (thus :  C^, 
Gi  D  A  e  hf,^  c4  g„^  d.J^  a^^  ej  ^4^  )•  ^^ow,  every  piano-key 
Avhich  sounds  b'^  has  to  serve  as  a  c-key  also:  but  b^^  is  too  sharp 
(has  too  many  vibrations)  to  give  a  perfect  Octave  of  C ;  hence, 
if  we  slightly  flat  every  Fifth  from  C  upward,  the  excessive  sharp- 
ness of  b^  is  sufficiently  corrected,  or  tempered,  to  enable  this  tone 
to  sound  the  perfect  Octave  required  {i.  e.,  the  tone  c),  while  re- 
maining M  sufficiently  exact  for  all  practical  purposes.  Moreover, 
as  the  same  c-key  has  to  serve  for  a  third  tone,  viz.:  d\^\f,  an  ap- 
proximation to  this  tone,  as  afforded  by  the  tone  c,  having  with  it 
almost  coincident  vibrations,  is  accepted  instead  of  d\f}f  mathomat- 
icallv  exact.     Thus  we  begin  to  see  how  35  tones  may  be  played 


*  This  flattinjr  of  the  Fifths  is  easily  tested  by  the(»xperimpnt  suggested  in 
the  Jif'iiiar/c  to  4(i  ;  the  g,  for  instance,  which  is  soun  led  by  the  correspond- 
ing? key  of  a  piano  properly  tuned,  will  be  perceived  by  .a  sensitive  ear  to  be 
somewhat  lower  than  the  over-tone  g  which  is  generated  by  the  string  c. 


36  MODERN     TONALITY. 

by  12  keys.  For  in  the  same  way  every  other  triplet  of  tones  which 
vibrate  almost  alike  is  reduced,  by  merging,  for  example,  d  \^  and  h  x 
into  (^,  e^  and  gl}l>  into  /, /x  and  «^^  into  g,  etc.,  etc.  Now,  by 
repeating  these  tones  in  a  higher  and  a  lower  pitch  we  obtain  a 
series  of  pei'fed  Octaves  of  each,  thus  of  Z-^,  d\^\^  and  c  simulta- 
neously, all  three  being  one  and  the  same  tone  as  to  sound — and 
80  on  of  the  rest  of  the  tones.  In  this  way  the  tuning  is — at  least, 
approximately — equally  tempered*  throughout  the  whole  key- 
board ;  and  although  the  Seconds,  Thirds,f  Fourths,  Fifths,  etc.^ 
lose  somewhat  of  their  mathematical  exactness  and  absolute  acous- 
tical purity,  yet  the  effect  is,  on  the  whole,  not  offensive  even  to 
the  most  sensitive  ear.  The  great  point  gained  is  the  absolutely 
necessary  series  of  perfect  Octaves.  Moreover,  modern  music  de- 
rives from  this  system  a  very  decided  advantage,  in  tlie  Enhar- 


*  We  shall  see  further  on  that  we  obtain  by  transposition  (see  Chap.  XIII) 
repetitions  of  the  model  scale  of  C,  each  one  in  a  dififerent  pitch — in  fact,  11 
additional  scales.  Now,  we  might  time  perfectly  all  the  intervals  in  one  par- 
ticular scale,  say  that  of  C ;  but  then  the  other  scales  would  be  out  of  tune, 
some  more,  others  less  so.  By  Equal  Temperament,  however,  the  various 
scales  are  tuned  alike — or  very  nearly  so,  the  acoustic  impurity  of  the  inter- 
vals (which  is  sometimes  called  the  "wolf")  being  equally  dispersed  through- 
out all  the  scales,  by  which  dilution  it  loses  its  offensiveness. 

f  Unlike  the  Fifths,  the  Thirds  must  be  tuned  slightly  sharp.  For 
instance,  I  tune  \.o  e  \is  perfect  major  Third,  e ;  then  to  e,  its  perfect  nuijor 
Third,  gdt ;  and  to  gt,  its  perfect  major  Third,  ht.  Now,  this  ht  has  to  serve 
also  for  c',  the  Octave  of  the  starting-tone,  c ;  moreover,  the  same  tone,  W,  has 
to  serve,  as  c',  for  the  major  Third  to  ah.  But  bt,  as  perfect  major  Third  to  gt, 
cannot  serve  as  c' .  Why  not  ?  Because,  c',  as  perfect  Octave  of  c.  vibrates 
twice  whilst  c  is  vibrating  once  (45),  whereas  ht  would  vibrate  too  slowly  for 
this.  But  we  must  absolutely  have  c'  exact  (perfect  Octave),  and  since  we  cannot 
get  this  by  three  ^e?'/«'^  major  Thirds  from  ^ ,  viz. :  e,  gt,ht,'we  must  needs 
sharp  each  Third  a  little.  Thus  the  major  Third  to  gt  will  actually  sound 
d  exact,  giving  the  desired  perfect  Octave  to  c,  and  at  the  same  time  the  tem- 
pered major  Third  (as  ht  slightly  sharped)  to  gt,  and  (as  c')  to  ab,  the  tone 
accepted  as  identical  with  gt. 


MODERX     TOXALITY.  37 

monic  notation  of  tones,  a  subject  to  be  considered  in  the  next 
chapter. 

Eemaek. — The  introduction  of  the  tempered  system  of  tuning 
dates  from  about  the  year  1700:  its  universal  acceptance  in  Ger- 
many is  ascribed  chiefly  to  the  exertions  of  the  organist  Werck- 
meister,  of  Halberstadt,  and  of  the  immortal  John  Sebastian  Bach. 
The  48  Piano  Prehides  and  Fugues  of  the  latter,  published  under 
the  title:  ''Das  wohltemperirte  Clavier"  (The  Well-tempered 
Clavichord),  were  composed  to  show  the  practicability  of  the  tem- 
pered system. 

QUESTIONS. 

43.  Are  all  the  Intervals  of  the  Scale  tuned,  in  keyed  instruments,  per- 
fectly, i.  €.,  with  mathematical  exactness  ?  What  is  the  technical  name  of 
the  modern  system  of  tuning?. . .  .44.  What  principle  has  been  followed  ia 
the  selection  of  the  13  tones  of  modern  music  ?  When  two  tones  form  aper- 
fect  Oct'ii:e,v>\iai  do  you  say  of  their  relationship?  What,  when  two  tones 
form  a  perfect  Fifth  ?  . .  .4o.  When  do  two  tones  form  a  perfect  Octave  ?. . . . 
40.  When  do  two  tones  form  a  perfect  Fifth  ?  How  can  we  hear  a  perfect 
Fifth  ?  (Remark.). . .  .47.  If  we  multiply  a  tone  by  the  perfect  Octave,  do  we 
obtain  new  tones,  strictly  speaking  ?  What,  if  we  multiply  a  tone  by  the  per- 
fect Fifth?. ..  .48.  What  do  we  lose  in  multiplying  atone  by  the  perfect 
Fifth?  Explain  this. . .  .41).  How  does  the  modern  system  of  tuning  meet 
this  difficulty  ?  Our  modern  tonal  system  consists,  then,  of  what  ?  By  what 
means  has  this  reduction  of  35  tones  to  12  been  effected?  What,  in  partic- 
ular, is  done  in  tuning  by  Fifths?  Give  an  example.  What  kind  of  Fifths, 
then,  exclusively,  will  admit  of  our  having  perfect  Octaves?  {Ans.  Tempered^ 
i.  e.,  slightly  flatted,  not  perfect  Fifths.)  When  two  tones — say  bi  and  dbb—- 
have,  according  to  acoustical  laws,  almost,  yet  not  exactly,  the  same  vibra 
tions  as  a  third  tone — say  c,  supposing  this  third  tone  already  tuned,  what  i- 
the  actual  process  of  reduction  by  which  the  3  tones  are  sounded  by  one  keyr 
{Ans.  Instead  of  It  and  dbb  exact  (implying  a  separate  key  for  each),  we  acc^p 
approximations,  suflBciently  near  for  practical  purposes,  i.  e.,  the  c-key  serves 
for  both  these  tones,  the  slight  discrepancies  of  vibration  being  ignored.) 
Give  instances  of  other  reductions  of  the  same  kind.  Does  oven  a  sensitive 
ear  require  that  all  the  intervals  be  tuned  with  absolute  (mathematical)  exact- 


38  MODERX     TONALITY. 

ness?  What  is  the  great  point  gained  by  the  tempered  system  of  tuning? 
What  other  advantage  does  modern  music  derive  from  this  system  ?  What 
is  the  date  of  the  general  introduction  of  this  system  ? 


CHAPTER    YII. 

The  Modern  Enharmonic  Scale. 

50.  In  comparing  the  descending  with  the  ascending  Chro- 
matic Scale  of  C  (Fig.  9,  p.  21)  we  notice  that  some  of  the  tones 
take  tivo  names.  Thus,  the  second  tone  ascending,  c^,  corresponds, 
in  sound,  exactly  with  the  last  but  one  descending,  yet  the  latter 
is  called  d\^.  Similarly,  d1^  in  the  ascending  series  appears  as  e\}  in 
the  descending;  again,  f^  in  the  ascending  as  g\^  in  the  descend- 
ing— and  so  on.  These  are  illustrations  of  tiie  effect  of  Equal 
Temperament,  as  explained  in  the  previous  chapter.  Strictly 
speaking,  c|  and  d\^  are  really  different  tones,  one  being  generated 
by  more  vibrations  than  the  other ;  and  the  same  is  to  be  said  of 
d^  and  e\^,  of  f^  and  ^[?,  and  so  on.  But,  as  we  have  seen,  the 
modern  practice  in  tuning  keyed  instruments  ignores  these 
mathematical  differences  (which  are,  after  all,  not  very  great),  and 
makes  one  single  sound  answer  for  three  sounds  with  different 
names,  while  yet  retaining  the  nmiies,  to  be  differently  applied  to 
one  and  the  same  tone,  as  the  case  may  require.  We  have  an 
analogy  to  this  in  spoken  language,  as,  for  .instance,  when  the  07ie 
sound  produced  in  pronouncing  "r-i-t"  (i  Jong)  is  variously  writ- 
ten "rite,"  "right,"  "write,"  "wright,"  according  as  different 
ideas  are  to  be  conveyed.  In  the  same  way.  one  and  the  same  tone 
will  have  to  be  written  in  various  notation,  according  to  circum- 
stances— and  this  is  what  constitutes  musical  oiihography,  the 
knowledge  of  which  is  so  necessary  to  the  musician.     For  a  few 


MODERN     TO  XA  L  I  T  T. 


39 


practical  hints  on  this  important  subject  we  refer  the  reader  to 
Appendix  II,  p.  86. 

51.  The  twelve  tones  of  our  modern  system,  written  in 
Tarious  notation,  form  what  is  called  the  Enharmonic  Scale,  an 
illustration  of  which,  derived  from  Fig.  9,  here  follows : 

Fig.  SI. 


Db 


Gb 


A 


B 


52.  In  the  above  figure  we  see  that  one  and  the  same  piano- 
key,  sounding  Cj  and  Ih,  gives  modifications  for  two  tones,  viz. : 
C'J  for  C,  D\^  for  I) — and  so  on  of  the  other  short  keys.  The  Enhar- 
monic Scale  above  illustrated  is,  however,  imperfect,  as  not  exhib- 
iting all  the  modijications  which  are  practicable  with  the  tones. 
For  instance  :  just  as  D'^  is  a  half-step  above  D,  and  A^  above  A,  so 
is  Fin  regard  to  E,  and  Cin  regard  to  B,  yet  this  figure  does  not 
relate  E  to  F,  nor  C  to  B.  Now,  according  to  the  theory  of  Enhar- 
monics,  each  and  every  tone  may,  under  certain  circumstances,  be 
regarded  as  a  modification  of  the  tone  on  the  decree  next  above,  or 
below.  The  tone  sounded  by  any  C-key  of  the  piano,  and  generally 
written  C,  may  therefore  sometimes  be  considered  a  raising  of  B 
(on  the  degree  below),  and  be  Avritten  B^.  for  it  is  the  only  B^  the 
piano  gives  us.  So,  too,  the  tone  sounded  by  any  5-key  will  some- 
times have  to  serve  as  CJ?,  and  Ije  written  accordingly;  ami  (he 
tone  generally  written  F  will  sometimes  have  to  be  Avritten  E^ ; 
and  similarly,  the  tone  E  will  sometimes  appear  as  F\y.  We  can 
therefore  enlarge  our  p]nharmonic  Scale,  as  follows  : 


40 


MODERN     TO  NA  LITY. 


Fig.  82. 


c« 

DJt 

Fjf 

GJ 

AS 

Db 

Eb 

Gb 

Ab 

Cb 

m 

Fb 

Elf 

Cb 

U 

c 

D 

E 

F 

G 

A 

B 

c 

53.  But  neither  does  Fig.  33  represent  a  complete  Enhar- 
monic Scale,  for  the  modification  of  a  tone  embraces  double  rais- 
ing and  double  depression  (see  18.  p.  19.),  implying  the  use  of  the 
K  and  \f\^.  In  fact,  it  often  happens  that  when  we  wish  to  raise  a 
tone  already  sharped,  or  to  lower  one  already  flatted,  musical 
orthography  requires  us  to  keep  the  tone  on  the  same  staff- 
degree,  and  to  use  the  x  or  [^[j.thus  raising  or  depressing  the  tone 
hy  two  half-steps.  In  this  way,  the  tone  whose  primary  name  is 
D  will  serve  as  C  double-sJmrp,  and  reciprocally,  the  primary  tone 
0  will  serve  as  D  double-flat.  So,  too,  JS  will  appear  as  the  double 
raising  of  Z)  (Z>  x  ),  and  Z)  as  the  double  depression  of  U  {^\}\}), 
F  as  G\^\^,  G  as  Fy.  and  as  A\^\^,  A  q,s,  Gx  and  as  B\^\^,  and 
5  as  yl  X.  Moreover,  among  the  intermediate  tones,  C^  will  ap- 
pear as  i?  X  ,  ^[7  as  F\^\^,  F^  as  Ex,  and  B\f  as  C\^\^.  The  complete 
Enharmonic  Scale  will  then  appear  as  follows: 


Fiff.  .5.5. 


Db 

Fbb 

Gb 

Ab 

Cb 

Eb 

F« 
Ex 

GS 

Bb 

Alt 

Dbb 

Ebb 

^M 

Fb 

Gbb 

1^           la- 
Abb 

m 

Cb 

Dbb 

C 

D 

E 

F 

G 

A 

B 

C 

Bit 

Cx 

Dx 

Elf 

Fx 

Gx 

Ax 

B« 

MODERN     TO  NA  LITY.  41 

54.  The  modern  Enliarmonic  theory  may  thus  be  summed  up : 
Each  tone  may  be  regarded,  1st,  ahsolutdy  (i.  e.,  without  relation 
to  the  tones  on  the  adjacent  degrees),  as  in  the  Diatonic  and  Chro- 
matic Scales ;  2d,  as  a  raising,  single  or  double,  of  the  tone  on  the 
degree  next  beloiv  ;  and  3d,  as  a  depression,  single  or  double,  of  the 
tone  on  the  degree  next  above.  Hence,  each  tone  (except  Gj^*}, 
takes  a  triple  name,  viz.  :  1st,  its  primari/  name,  either  simple  or 
with  a  qualification  added  ;  Jd,  the  name  of  the  degree  next  beloiv, 
properly  qualified  ;  and  Sd,  the  name  of  the  degree  tiext  above, 
properly  qualified. 

55.  The  distinguishing  characteristics  of  the  Diatonic,  the 
Chromatic,  and  the  Enharmonic  notation,  may  be  thus  abbrevi- 
ated, for  memorizing : 

DiATOXic :  difierent  tones,  names,  and  seats  on  the  staff. 
Chromatic  :  alteration  of  a  tone,  same  name  and  seat  on  the 

staff. 
ExHARMOXic :  one  and  the  same  tone,  with  three-fold  name 

and  seat  on  the  staff. 

56.  When  one  and  the  same  tone  is  repeated,  the  second  time 
in  a  different  notation  (as  frequently  occurs),  this  is  called  an 
Enharmonic  change,  as  in  the  following  example  : 


*  Why  (t  $  is  alone  excepted  is  very  easily  exjJained.  Every  tone  can  be 
related  to  (regarded  as  a  modification  of)  a  primary  tone  distant  a  half  step,  or 
even  two  half-steps,  hut  not  farther.  Tlius — to  compare  the  other  altered  tones 
with  Gt — the  tone  C%  is  not  only  a  single  depression  of  B  and  raising  of  C, 
but  also  a  double-raising  of  the  tone  two  half-steps  distant,  viz.,  B ;  similarly, 
Di  is  related,  not  only  to  D  and  E,  but  also  to  F ;  Ft,  not  only  to  F  and  G, 
but  also  to  E ;  and  Ai,  not  only  to  A  and  B,  but  also  to  C.  But  Gt  cannot 
be  related  to  the  primary  tone  F,  for  tliis  tone  lies  three  half-steps  distant,  nor 
to  the  primary  tone  B,  for  the  same  reason.  Hence,  Gt  can  be  related  only 
to  the  primary  tones  G  and  A,  i.  e.,  as  the  single  raising  of  the  former, 
and  single  depression  of  the  latter ;  it  therefore  cannot  have  more  than  two 
names. 


42  MODERN     TONALITY. 


Fig,  34. 


53: 


Ell.  Change. 

57.  It  should  be  added,  in  concluding  this  chapter,  that  in  our 
modern  tempered  system  of  tuning,  the  real  significance  of  the 
word  "  enharmonic,"  as  employed  in  ancient  times,  is  lost ;   for  a 

genuine  enharmonic  scale  would  be,  for  instance,  c,  c^,  d^,  d,  d^,  e\^, 

e,  4  /,  yj,  (j]^,  (J,  y%  «l7,  a,  a^,  Z*^,  h,  l'^,  c'—m  all,  19  diferetit 
tones ;  whereas,  by  considering  every  pair  of  tones  thus  connected 
- — ■  as  one  and  the  mine  tone  variously  named,  as  in  our  modern 
system,  we  have  only  12  different  tones,  not  counting  the  Octave. 
Hence  the  Enharmonic  Scale  proper"  has  disappeared,  being  re- 
duced to  our  Chromatic  Scale,  and  the  expression  "enharmonic" 
denotes  merely  a  varying  notation  for  one  and  the  same  sound. 

QUESTIONS. 

50.  What  illustrations  of  Equal  Temperament  do  we  find  in  Fig.  9, 
p.  21  ?  Exjilain  how  it  is  that  ct  and  ch,  for  instance,  are  played  by  one  and 
the  same  key.  Is  there  anything  in  spoken  language  analogous  to  the  princi- 
ple of  making  one  musical   sound  serve  for  three  tones  ?     For  instance . 

In  what  does  "  musical  orthograi)hy  "  consist?.  .  .  .51.  What  is  the  so-called 
"Enharmonic"  Scale?  .  5'2.  Explain  Fig.  31,  and  show  why  it  does  not 
represent  a  complete  Enharmonic  Scale. . .  .53.  Explain  why  the  Enharmonic 
Scale  represented  by  Fig.  33  is  not  yet  complete. . .  .5-4.  Give  a  summary  of 
the  modern  Enharmonic  theory.  Why  does  Ot  alone  take  hut  one  additional 
name  {Ah)t  {Ans.  Oi  could  not  be  related  to  F  but  by  being  written  Fx 
raised  {FW),  nor  to  B,  but  by  being  written  Bbh  lowered  {Bbhb).  Now,  we  do 
not  go  further  than  double  raising  or  depression,  in  modifying  a  tone  ;  hence, 

Gt  can  be  related  only  to  O  and  A.     See  JS/ote.) 55.  What  are,  in  a  few 

words,  the  cliaracteristics  of  Diatonic  notation  ?     Of  Chromatic  ?     Of  Eriha/r- 

monicf. .  .  .5(i.  What  is  the  "  Enharmonic  change"? 57.  What  is  the 

difference  between  the  ancient  Enharmonic  system  and  the  modern  ?  What 
is  the  real  significance  of  the  word  "  enharmonic"  in  modern  music? 


MODERy     TONALITY,  43 


CHAPTER    VIII. 

Modification  of  Intervals  by  Chromatic  Alteration. 

58.  As  we  have  already  seen,  the  tones  of  the  Diatonic  Scalft 
may  be  modified,  i.  e.,  chromatically  altered;  from  this  follows  the 
practicability  of  modifying  the  Irdervals  of  the  Diatonic  Scale. 

59.  It  must,  however,  be  well  understood  that  in  the  case  of 
the  Interval,  implying  a  relationship  befiveen  two  tones,  chromatic 
alteration  in  the  broad  sense  does  not  necessarily  involve  wluit  we 
mean  by  modification,  i.  e.,  a  change  in  the  nature  of  the  interval 
— it  may  effect  only  transposition.  In  Fig.  9,  page  21,  are  several 
instances  of  chromatic  alteration,  e.  g.  the  Fifth,  c-g,  into  (-'^-g'^y 
the  Fourth,  d-g,  into  d'^-g^^  etc.,  yet  in  each  case  the  nature  of  the 
interval  is  unchanged,  (^-g^^  for  instance,  forming  the  same  bind 
of  Fifth  (major)  as  c-g.  What  has  taken  place  is  not  exactly 
chromatic  alteration  but  rather  Transposition,  wliich,  as  applied  to 
an  interval,  changes  the  pitch  of  both  tones  equally  and  in  the  same 
direction,  i.  e.,  raises  or  lowers  each  tone  by  the  same  number  of 
half-steps,  steps,  or  degrees,  so  that  the  interval  remains  the  same 
in  kind  as  V)efore  transposition. 

00.  Different  from  the  effect  of  trans]iosition  is  that  of  chro- 
matic alteration  in  its  usual  and  strict  sense,  upon  an  interval — it 
changes  the  interval  -as  to  kind,  and  in  some  cases  even  creates 
additional  kindx,  besides  the  two  normal  kinds — minor  and  major 
— found  in  the  Diatonic  Scale. 

f»l.  This  chromatic  alteration  or  modification  of  intervals  is 
limited  in  extent  to  a  chromatic  half -step,  and  it  effects,  1st,  the- 
change  of  minor  intervals  into  major,  and  contrariwise  ;  or,  3d, 
the  diminution  of  minor  intervals;  or,  3d,  the  avgmentation  of 
major  intervals.  The  latter  two  processes  give  rise  to  two  new 
kinds  of  intervals. 


44  MODERN     TONALITY. 


Minor  Intervals  changed  into  Major,  and  contrariwise. 

G2.  Since  the  Formula  of  a  majo7'  interval  invariably  gives 
one  chromatic  half-step  more  than  that  of  a  minor-  interval  of  the 
same  denomination  (see  Figs.  21 — 30,  pp.  31,  32),  it  follows  that 
to  change  a  minor  into  a  major  interval  we  have  only  to  make  the 
former  greater  by  such  a  half-step,  which  is  done  by  chromatically 
raising  the  tipper  tone  or  lotveritig  the  loiver  tone.  In  cbanging, 
•on  the  contrary,  a  major  into  a  minor  interval,  the  upper  tone  is 
chromatically  loivered,  or  the  lower  one  is  chromatically  raised. 

Remark. — The  chromatic  alteration  whereby  minor  intervals 
are  changed  into  major,  and  contrariwise,  effects  the  transposition 
of  intervals,  in  this  sense,  that  every  pair  of  staff-degrees  which  is 
the  normal  seat  of  a  minor  interval — for  instance,  a-c  (minor 
Third),  e-a  (minor  Fourth),  h-f  (minor  Fifth),  etc. — may  be  used 
also  for  a  major  interval  of  the  same  denomination,  whereby  major 
intervals  may  be  said  to  be  transposed  to  new  seats  on  the  staff. 
But,  while  the  chromatic  alteration  of  minor  into  major,  and  con- 
trariwise, accidentally  effects  transposition  in  this  way,  yet  its 
primary  effect  is  to  change  an  interval  as  to  kind,  Avhich  is  not 
done  in  transposition  proper,  as  we  have  seen. 

Exercise.* 

Change  all  the  intervals  of  the  Diatonic  Scale  of  C  (from  c  to  a, 
inclusive),  in  the  following  manner: 

1.  Every  minor  into  major,  in  both  ways. 

2.  Every  inajor  into  minor,  in  both  ways. 


*  In  writing  this  and  the  subsequent  exercises,  it  would  be  well  to  leave  a 
blank  staflF  under  that  on  which  the  exercise  is  written,  for  such  corrections 
■as  may  be  necessary,  so  that  these  may  stand  immediately  under  their  several 
corrigenda. 


M  0  D  E  R  y     TO  XA  L  IT  Y.  45 


Diminution  of  Minor  Intervals, 

63.  To  diminish  a  minor  interval  (make  it  less  than  minor), 
is  to  bring  the  two  tones  nearer  together  by  a  chromatic  half-step. 
This  is  efiFected  by  chromatically  lowering  the  upper  or  raising  the 
lower  tone. 

Fiff.  33. 

Minor  Third.  Diminished  Third. 


-=p— :>- 


idra; 


1 


Remark  1. — The  Formula  of  the  diminished  Third  is  the 
same  as  that  of  the  major  Second,  viz.:  3  (see  Fig.  20,  ji.  31  .  But 
the  two  intervals,  having  a  different  harmonic  and  melodic  sig- 
nificance, are  differently  written.  It  will  be  noticed  that  whereas 
the  major  Second  is  formed  by  a  cliromatic  and  a  diatonic  half- 
step,  the  diminished  Third  is  formed  by  two  diatonic  half-steps. 

Remakk  2. — Since  to  diminish,  in  the  present  sense,  is  to  ren- 
der a  minor  interval  smaller,  it  follows,  1st,  that  there  cannot  be  a 
diminished  Second^  the  minor  Second  being  the  smallest  interval 
used  ;  and  2d,  that  the  term  "  diminished  "  should  be  used  oid y  of 
an  interval  proximately  derived — by  chromatic  cdteration — from  a 
minor  interval  of  the  same  denomination.  An  illogical  exception  is^ 
however,  commonly  made  of  the  small  Fifth — as,  b-f,  d-ak,  which 
is  called  "diminished."  Now  the  Fifth  of  this  construction  (see 
36,  p.  28)  is  a  normal  interval  in  a  diatonic  scale,  and  never 
appears  as  a  minor  Fifth  made  smaller.  A  (jenuine  diminished 
Fifth,  on  the  other  hand,  is  essentially  a  chromatically  uttered 
Fifth,  and  represents  the  interval  in  its  smallest  possible  measure- 
ment, as,  for  instance,  s  J,  equal  to  a  mino'*  Fourth. 


46-  MODERN     TO  NA  L  I  T  Y. 


Augmentation  of  Major  Intervals. 

64.  Augmentation  is  the  contrary  of  diminution — to  augment 
an  interval  which  is  by  supposition  already  major,  is  to  make  it 
still  greater  by  a  chromatic  half -step. 

65.  Augmentation,  like  diminution,  may  be  practised  in  two 
ways ;  but,  unlike  diminution,  is  effected  by  chromatically  raising 
the  upper  or  loivering  the  loiver  tone. 

Fiff.  30. 

Major  Second.  Augmented  Second. 


\=^^      t»'=* — g='^z=[|:rfez^^^ba^:r:g 


13     3  12      3       4  1       3       3       4 

Remark  1. — The  Formula  of  the  augmented  Second  is  the 
same  as  that  of  the  viinor  Third,  viz. :  4  (see  Fig.  21,  p.  31).  But, 
as  in  other  similar  cases,  the  two  intervals  have  a  different  har- 
monic and  melodic  significance,  and  are  therefore  differently  writ- 
ten. The  minor  Third  is  formed  by  two  diatonic  half-steps  and 
one  chromatic,  the  augmented  Second  by  tivo  chromatic  half-steps 
and  one  diatonic. 

Remark  3. — To  augment,  in  the  present  sense,  is  to  render  a 
major  interval  greater.  Hence,  the  common  practice  of  calling  a 
Fourth  like/-6  "augmented  "  is  inconsistent  and  confusing.  For 
the  Fourth  of  this  construction  (see  35,  p.  28)  is  normal  in  the 
diatonic  scale,  and  never  appears  as  a  major  Fourth  enlarged. 
Wliereas,  a  genuine  augmented  Fourth  is  essentially  a  chromati- 
callg  altered  Fourth,  and  represents  the  utmost  possible  extension 

of  the  interval,  as,  for  instance,  ]t,  equal  to  a  major  Fifth. 

66.  From  what  has  been  said  in  this  chapter  we  sum  up  the 
following  :  \st.  That  the  chromatic  alteration  of  intervals  in  the 
Diatonic  Scale  impl'es  either  simple  transposition,  or  modification 


MODERN     TO  KA  L  I  T  Y.  47 

— chromatic  alteration  in  the  usual  and  strict  sense — the  latter 
changing  the  interval  as  to  kind,  Avhich  the  former  does  nt>t.  2d. 
That  anv  minor  interval  of  the  Diatonic  Scale  may  be  chromat- 
ically changed  into  major,  and  contrariioise.  Sd.  That  some  minor 
(and  only  minor)  intervals  may  be  diminished,  by  bringing  the  two 
tones  nearer  together  by  a  chromatic  half-step.  4fk-  That  .-ome 
major  (and  oiili/  major)  intervals  may  be  augmented,  by  puttin^'j  an 
additional  (chromatic)  half -step  between  the  two  tones. 

Half-step  Formulas  of  all  the  Intervals,  including  those  ch/i- 

maticallv  altered. 

The  Formula  of  the  Minor  Second  is  3. 
Major  Second  is  3. 
Augmented  Second  is  4- 
Dwiinished  Third  is  3. 
Minor  Third  is  4. 
Major  Third  is  5. 
Augmented  Third  is  6. 
Diminished  Fourth  is  5, 
Minor  Fourth  is  6. 
Mujoi'  Fourth  is  7. 
Augmented  Fourth  is  8. 
Diminished  Fifth  is  G. 
Minor  Fifth  is  7. 
Major  Fifth  is  S. 
Augm,ented  Fifth  is  9. 
Diminished  Sixth  is  S. 
Minor  Sixth  is  9. 
Major  Sixth  is  10. 
Augmented  Sixth  is  11. 
Diminished  Seventh  is  10, 
Minor  Seventh  is  11. 
Major  Seventh  is  IB. 


48  3I0DERN      TOXALITT, 


Exercises. 

1.  Diminish,  in  both  ways,  the  following  minor  intei'vals : 
Thirds,  cl-f,  5-^,  c^-e,  h\l-g,  e-g,  a-c,  /||-«,  <^i^-f- — Fourths,  g-c, 
e-a,  /If-^,  g-d,  c-f,  a-d. — Fifths,  d-a\z,  b-f,  a^'e,  g^-d,  «-el2.— 
Sixths,  c-e,  J'-a,  d-F^,  ^-b. — Sevenths,  d-c,  A-g,  <-'!^-b,  e-d,  f-  Gy 
a-b,  c-V^. 

2.  Augment,  in  both  ways,  the  following  major  intervals : 
Seconds,  d-e,  ^-f,  f-g,  Th-c,  c-d,  e-f^,  a-b,  g-a. —  Thirds,  c-e, 
«-/ft  ^-^)  f-a.— Fourths,  Ak-d,  J-B,  e-c/jf,  d-y^,  e^-a.— Fifths, 
G-d,  d-a,  c-g,  f^c^,  e-b,  f-c,  e]z-b\i,  (lf-^|f- — Sixths,  d-b,  f-d,  c-a, 

«-/ft  ^i2-^>  ^-ej,  e\z-c. 

QUESTIONS. 

58.  Are  the  Intervals,  as  found  in  the  Diatonic  Scale,  unchangeable  ? . . . , 
59.  Can  an  interval  be  chromatically  altered  without  being  modijied? 
What  is  the  effect  of  Transposition  on  an  interval  ?. . .  .OO.  What  effect  has 
chromatic  alteration,  in  its  usual  sense,  on  an  interval  ?. . .  CI .  What  is  the 
limit  of  the  chromatic  alteration  or  modification  of  an  interval  ?  What  are 
the  effects?  ..  (i'2.  How  is  a  minor  changed  into  a  major  interval?  A  major 
into  a  minor  interval  ?  How  does  this  modification  accidentally  effect  the 
transposition  of  intervals?  (Remnrl:). . .  Aiii.  What  is  it,  to  diminish  a 
minor  interval  ?  How  is  it  done  ?  What .  minor  interval  is  never  dimin- 
ished ?  Give  the  reason. — To  what  species  of  interval  should  we  exclu- 
sively apply  the  term  "diminished"?  What  do  you  say  of  the  usual 
practice  of  calling  the  minor  Fifth  "diminished"?  What  is  a  genuine 
diminished  Fifth?  {Bemark  -?.)  .  -  64.  What  is  it  to  augment  a  major  inter- 
ral  ? . . . .  05.  How  is  augmentation  effected  ? — What  do  you  say  of  the 
tommon  practice  of  calling  the  major  Fourth  "  augmented  "  ?  Explain  this 
tiatter,  and  state  what  is  a  genuine  augmented  Fourth.  {Bemark  2). . .  .66* 
Give  a  summary  of  this  chapter. 


MODERN    TONALITY. 


49 


CHAPTER     IX. 

Inversion  of  Intervals.* 

67.  To  invert  an  interval  is  to  transpose  one  of  its  two  tones 
by  the  Octave,  the  result  being  that  the  same  two  tones  form  an- 
other intei'val  as  to  denomination  and  hind.  Simple  change  in 
the  order  of  the  two  tones,  as  in  Fig.  37,  does  not  effect  inversion, 
but  merely  makes  a  descending  out  of  an  ascending  interval,  or 
vice-versa. 

Fiff.  37. 


1      6 


1      6 


68.  The  rule  of  inversion  is  this:  transpose  the  lotver  tone  an 
Octave\  higher,  or  the  upper  tone  an  Octavef  lower,  &o  that  the 
tone  which  was  the  upper  is  now  the  lower.  The  following  are, 
examples : 

Fiff.  :is. 

Ma.j.  Sccniid:      inverted.         Mill.  Third:      inverted. 


Dim.  Third:      inverted. 


13         16 

Aii,s,nn.  P'ifth;    inverted. 


16  15 


:|« 


*  The  Inversion  of  intervals  is  in  some  works  improperly  styled  the 
"Transposition"  of  intervals.  But  w.o  have  seen  that  transposing  doos  not 
affect  the  7mti'rr.  of  th:  intcrml  as  such,  whereas  Inversion  does. 

•f-  Or  two  Octaves,  three  Octaves,  etc  ,  higher  or  lower— the  number  of  Oc- 
taves does  not  in  the  least  affect  the  inversion.     See  first  Note,  p.  24. 
3 


50  MODERN     TONALITY. 

69.  From  the  foregoing  examples  we  see  that  the  jnajor  Second 
c'-d'  is  changed  into  the  minor  Seventh  d'-c",  the  minor  Third 
e'-g  into  the  major  Sixth  g'-e",  the  diminixlied  Third  d\-f'  into 
the  augmented  Sixth  f'-d  '^,  and  the  augmented  Fifth  f  -c"|  into 
the  diminished  Fourth  c'^/"-  Inversion,  therefore,  elfects  the 
chan2"3  of  an  interval  both  in  denomination  and  in  kind. 

70.  The  changes  of  denomitiation  are  expressed  in  the  formula: 

7   P)   T  4  S   2 

r,  o   A   ►  ^  ^ ,  which  may  be  read  thus,  from  the  lower  row  of 

figures  upward:    the  Second,  inverted,  becomes  a  Seventh;    the 
Third,  a  Sixth — and  so  on  of  the  rest. 

71.  The  changes  of  kitid  effected  by  the  inversion  of  intervals 
are  thus  summed  up:  minor  becomes  major,  Siud  cotitrariwise ; 
diminished  becomes  augmented,  and  contrariioise. 

Exercise. 

Invert,  giving  two  or  three  examples  of  each,  all  the  intervals 
(a  full  enumeration  of  which  is  given  under  the  heading,  "Half- 
step  Formulas,"  etc.,  p.  4'7),  for  instance,  the  minor  Seconds,  e-f, 
d!-e\,  d\-e;  the  major  Seconds,  A-B,  d''^-e'\^,  B-c^ ;  the  aug- 
mented Seconds,  F-G^,  c-d^,  d'\^-e,  etc.,  etc. 

QUESTIONS. 

67.  What  is  it,  to  invert  an  interval?    Wliat   is  tlie  result?    What  is 

the  effect  of  merely  changing  the  onhr  of  tlie  two  tones  of  an  interval  ? 

68.  State  the  rule  of  inversion.     (Does  the  expression  "an  Octave  higher" 
or  "an  Octave  lower"  necessarily  mean  only  one  Octave  higher   or  lower? 

gee  Note  2.) 69.  What   double   change  in   an   interval   does   inversion 

effect  ? 70.  State  the  changes  of  denomination 71.  State  the  changes 

of  kind. 


MODERN     TO  N  A.  L  IT  Y.  51 


CHAPTEE    X. 

Intervals   as    Symphones.— Consonances   and    Disso- 
nances. 

72.  We  shall  now  consider  the  interval  from  that  point  of 
view  which  supposes  its  two  tones  to  be  sounded  simultaneously. 

73.  A  simultaneous  sound  of  two  or  more  tones  constitutes  a 
Symphoxe  (German,  '-der  Zusammenklang").*  We  are  at 
present  concerned  with  the  two-voiced]  Symphone,  in  other  words, 
that  formed  by  the  two  tones  of  an  interval,  sounded  together.  It 
is  classed  as  either  a  Co:n^soi^axce  or  a  Dissoxance.  Each  of 
these  terms  has  relation  not  exactly  to  the  absolute  sound  of  the 
Symphone,  considered  as  agreeable  or  disagreeable,  but  rather  to 
the  necessity  or  non-necessity  of  its  connexion  with  another  Sym- 
phone. 

74.  A  Consonance  is  a  Symphone  which  is  self-sufficient  and 
independent ;  either  of  its  tones  may  progress  to  another  tone,  but 

*  We  have  absolutely  no  word  in  oyr  language  corresponding  to  the  gen- 
eric German  word  "  Zusammenldang  "  (literally,  a  "  together-sound  "),  except 
"  Consonance,"  which,  indeed,  expresses  it  exactly,  but  is  used  in  a  specijie 
sense  only.  I  have  therefore  ventured  to  introduce  the  new  word  "  Sym- 
phone," derived  from  the  Greek.  The  expression  is  a  comprehensive  one, 
including  what  are  called  "Chords,"  "Triads,"  "Consonances,"  "Disso- 
nances," etc., — in  short,  any  combination,  agreeable  or  otherwise,  of  musical 
tones.  The  Interval  has  been  hitherto  considered  from  a  melodic  stand-point 
and  indicated  horizontally  (so  to  speak),  as  for  instance,  c-c,  c-g :  regarded  as 
a  Symptione,  in  other  words,  from  a  harmonic  stand-point,  it  is  indicated  ver- 
tically, for  example '-.,.,  etc. 
c     c 

f  The  expression  "  voice  "  is  often  used  as  synonymous  with  "  tone,"  irre- 
spectively of  the  tone  producing  agency.  Thus,  a  "  two-voiced"  Symphone  is 
one  composed  of  two  tones,  whether  sung  or  played. 


52 


MODERN     TONALITY. 


need  not,  as  far  as  any  incompleteness  of  the  Symphone  is  con- 
cerned. 

75.  A  Dissotiance,  on  the  contrary,  is  a  symphone  essentially 
transitory  in  character,  pointing  to  what  is  called  its  resolution 
into  a  consonance,  by  the  2Jrogression  of  one  or  both  of  its  tones. 
The  resolntion  need  not  follow  at  once — a  dissonance  may  lead 
into  another,  and  so  on,  leaving  only  the  last  one  to  be  really  re- 
solved into  a  consonance. 


Fig.  5.9.— Consonances. 
a  be  d        e  f 


1^=^ 


ff 


C) 


Fig.  40. — Dissonances. 


etc. 


76.  Each  of  the  Consonances,  as  in  Fig.  39,  appears  as  an  in- 
dependent symphone,  not  pointing  to  another  as  its  complement. 
The  Dissonances  (Fig.  40),  on  the  contrary,  severally  tend  to 
their  resolutions,  which  are  indicated  in  small  notes. 

77.  Some  dissonances— viz. :  the  harshest — reqnire  what  is 
called  preimration  also ;  i.  e.,  just  as  they  must  resolve  into  con- 
sonances, they  must  likewise,  as  it  were,  grow  out  of  preceding 
symphones,  so  as  not  to  be  introduced  too  abruptly.*  Thus,  a 
dissonance  is  said  to  be  "  prepared  "  if  one  of  its  tones  was  heard 
in    the   symphone    immediately  before.     In  the    following    Fig- 

f 
ure,  at  A,  the  harshness  of  the  dissonance  J ,    is  softened  by  the  e 

having  been  heard  in  the  previous  consonance  ^,  (as  may  be  proved 


*  Of  course,  a  composer  is  perfectly  free  to  introduce  a  dissonance 
abruptly  in  order  to  produce  some  particular  effect.  In  general,  the  rule  of 
preparation  of  dissonances  is  not  now  so  strictly  adhered  to  as  formerly. 


MODERN     T  O  XA  L  IT  Y.  53 

by  taking,  against  g',  some  other   tone  than  e',  as  in  the  same 

Figure,  at  B) :   the  dissonance,  thus  prepared,  then  resolves  inl  ♦ 

f 
the  consonance  -S/  . 

Fig.  41. 
A  B 


P 


1^ 


:j=d=e=t^=, 


I 


78.  The  difference  between  the  Consonance  and  the  Disso- 
nance may,  for  practical  purposes,  be  stated  thus:  the  Consonance 
may  be  introduced  abruptly,  and  may  cease  sounding  as  abruptly, 
or  may  l'»e  indefinitely  repeated,  or  may  pass  to  another  consonance 
or  to  a  dissonance.  The  Dissonance  is  not  generally  introduced 
abruptl}',  and,  once  introduced,  is  not  perfectly  free  as  to  its  dis- 
continuance or  its  next  progression,  as  has  been  explained. 

79.  The  Consonances  are  the  following,  an  example  of  each 
being  given  in  Fig.  S9 :  the  Octave  (a) ;  the  major  Fifth*  (b),  and 
its  inversion,  the  ininor  Fourth*  (c) — the  latter  is,  however,  under 

*  It  has  already  been  stated  that  in  many  works  the  minor  Fourth  is  called 
the  "  perfect "  Fourth,  and  the  major  Fifth  the  "  perfect  "  Fifth.  We  may  now- 
state  that  the  expression  "  perfect "  is  used  to  mark  a  peculiarity  of  tbese  two 
denominations  of  intervals  as  distinguished  from  the  other  consonant  inter- 
vals, viz. :  tluit  the  former  cannot  be  altend  without  ceasing  to  be  conso- 
nances. The  Tlnrd  and  the  Sixtti  may  be  changed  from  major  into  minor, 
and  contrariwise,  and   yet  remain  consonances  ;  but  if  the  Fourth — for  in- 

r' 
Stance,     ,  be  made  major,  or  otherwise  altered,  it  becomes  dissonant,  so  ajso 

the  Fifth — say  ^.  ,  if  made  minor,  or  otherwise  altered.     Hence  the  predicate 

•'perfect"  is  applied  to  the  minor  Fourth  and  the  major  Fifth,  to  denote  that 
single  condition  in  which  they  are  consonances.  Nevertheless,  many  of  the 
best  modern  musical  theorists  use  the  expressions  "minor "and  "major" 
instead,  deeming  the  significance  of  the  word  "perfect,"  as  just  explained, 
not  important  enough  to  justify  an  exception  of  the  Fourth  and  the  FiftJi  in 
that  simjjle  and  consistent  system  wliich  classes  the  normal  intervals  as 
minor  or  major,  according  to  their  measurement. 


54  MODERN     TONALITY. 

certain  circumstances  a  dissonance  (as  explained  in  the  study  of 
Harmony)  ;  the  mcyor  Third  (d),  with  its  invereion,  the  tninor 
Sixth  (e)  ;  and  the  minor  Third  (/) ,  with  its  inversion,  the  major 
Sixth  {(/).  All  the  other  intervals  are  dissonances,  viz.  :  the 
Second,  of  any  species  ;  the  diminished  and  augmented  Tliird ; 
the  diminished,  augmented,  major,  and  sometimes  the  minor. 
Fourth  ;  the  diminished,  augmented,  and  minor  Fifth ;  the 
diminished  and  augmented  Sixth  ;  and  the  Seventh,  of  any  species. 
(See  Appendix  III,  p.  91.) 

80.  Some  dissonances  do  not,  of  themselves,  {.  e.,  without  the 
addition  of  a  third  voice,  strike  the  ear  as  dissonant,  in  the  sense 
already  explained.  This  is  the  case  with  some  of  those  chromat- 
ically altered  intervals  whose  formulas  severally  coincide  wi;h 
those  of  consonances  (see  p.  47).  Thus,  the  dissonant  angmented 
Second  sounds  exactly  like  the  consonant  minor  Third  (for  instance, 

^v  like  J  J ),  the  dissonant  diminished  Fourth  like  the  consonant 
major  Tliird  (for  instance,  ^^^  like  ^*'\  etc.,  etc.  Other  disso- 
nances, on  the  contrary,  at  once  strike  the  ear  as  such — a  progres- 
sion of  one  of  the"  tones  seems  necessary,  if  the  ear  is  to  be  satis- 
fied. Such  dissonances  are  the  minor  Second,  the  major  Second^ 
the  diminished  Third,  the  major  Fourth,  the  minor  Fifth,  the 
awpnented  Sixth,  the  ininor  Seventh,  and  the  major  Seventh. 

81.  It  should  be  Avell  understood  that  although  extremely 
harsh  dissonant  combinations  are  possible,  yet  the  expression 
"dissonance"  does  not  necessarily  or  invariably  imply  harshness. 
Some  of  the  most  charming  effects  in  harmony  are  produced  by 
means  of  certain  dissonant  combinations,  and  it  might  surprise 
one  not  learned  in  music  to  be  told,  after  listening  with  deliglit  to 
the  performance  of  a  composition,  that  the  composition  abounded 
in  dissonances,  or,  as  some  say,  "discords."  Indeed,  without  the 
intermingling  of  dissonances  a  succession  of  consonances  would 
soon  be  found  insipid,  just  as  repose  is  not  thoroughly  enjoyable 
except  after  exertion.    Now,  as  Dr.  Macfarren  observes,  the  disso-* 


MODERN     TO  NA  L  IT  T.  55 

nance,  as  being  inconclusive,  and,  however  prolonged,  requiring 
resolution,  "is  thus  the  musical  exponent  of  unrest,  activity, 
aspiration."  Hence  the  best  preparation  for  the  repose  expressed 
by  the  consonance  is  the  activity  typitied  by  the  dissonance.* 

QUESTIONS. 

73.  "What  is  a  Symphone  ?  How  is  the  two-voiced  Symplioue  classed  ? 
"What  does  each  expression  properly  signify  ?. . . .  74.  What  is  a  Consonance  ? 
. . .  c  75.  What  is  a  Dissonance  ?  Must  a  dissonance  always  be  resolved  into  a 
consonance  in  the  follomng  progression  V. . .  .77.  What  is  generally  required 
in  the  case  of  the  harshest  dissonances  ?  When  is  a  dissonance  prepared  ? 
(Is  this  preparation  always  and  absolutely  necessary?  See  Note!).... 
78.  State  the  practical  dilFerence  between  the  Consonance  and  the  Disso- 
nance.... 71).  Name  the  Consonances.  (Why  is  the  expression  "perfect'* 
sometimes  applied  to  the  minor  Fourth  and  the  major  Fifth  f  See  Note.) 
Name  tlie  Dissonances. ..  .80.  Do  all  the  dissonances,  of  themselves,  strike 
the  ear  as  such  ?  Which  are  those  that  do  not  ?  Which  are  tliose  that  do  ? 
....81.  Does  the  term  "  dissonance  "  necessarily  imply  harshness  ?  What 
is  the  practical  use  of  dissonances  in  a  piece  of  music  ? 

*  An  acoustical  phenomenon  in  connection  with  this  subject  is  thus 
described  by  Professor  Macfarren  in  his  "  Lectures "  :  "Vibrations  are  more 
or  less  rapid  in  proportion  as  the  sound  is  higher  or  lower  of  which  they  are 
the  utterance,  and  combinations  (of  tones)  are  more  or  less  conmnant  in  pro- 
portion to  the  greater  or  less  number  of  coincident  vibrations  of  the  two 
sounds — thus,  two  sounds  in  unison  (the  most  perfect  consonants)  vibrate 
simultaneously ;  the  upper  note  of  an  8th  has  two  vibrations  for  every  one 
of  the  lower  ;  while  the  major  Seventh  (one  of  the  harshest  discords)  has  fif- 
teen vibrations  of  its  upper  note  for  every  eight  of  its  lower,  the  coincidences 
of  which  vibrations  are  as  rare  as  the  dissonance  of  the  combination  is  ob- 
vious." 


56  MODERN    TONALITY. 


CHAPTER    XI. 

The  Triad,  or  Three-voiced  Chord. 

83.  Under  the  general  term  "  symplione  "  is  included  what  is 
called  the  Chord,  which  is  a  simultaneous  sound  of  several  tones* 
combined  according  to  certain  laws.  The  particular  species  of 
chord  which  we  must  now  discuss — though  only  at  such  length  as 
is  necessary  for  the  purposes  of  this  work — is  that  which  is  called 
the  Triad. 

83.  The  Triad  is  a  symphoue  of  three  tones,  so  notated  that 
the  uppermost  tone  is  a  Fifth  of  some  kind,  and  the  middle  tone 
a  Third  of  some  kind,  to  the  lowest  tone,  or  Fundamental.  In 
other  words,  a  Triad  consists  of  a  tone  assumed  as  a  fundamental, 
with  its  Third  and  Fifth  above,  sounding  together  with  it,  as  in 
the  following  Figure : 

Fi(j.  4:2. 

a  h         c 


g=pEj=iOl 


84.  The  Triads  in  general  are  distinguished  by  the  hind  of 
Third  only,  or  by  the  ]ci7id  of  Fifth  only,  or  by  the  kind  of  Third 
and  of  Fifth,  peculiar  to  each. 

85.  In  Fig.  Jf2,  at  a  and  at  h  the  Fifth  is  in  both  cases  major, 
while  the  Third  at  a  is  major,  and  at  I  is  7ninor.  The  Triad  at  a 
is  a  major  Triad,  that  at  ^»  is  a  minor  Triad.  Each  takes  its  dis- 
tinctive name  from  its  different  kind  of  Third  only,  the  same  kind 
of  Fifth  (major)  being  common  to  both. 

*  At  least  three  different  tones  are  requisite  to  form   a  complete  cliord. 
Hence,  certain  two-voiced  combinations  are  chords  in  a  limited  sense  only. 


MODERN     TOXALITY.  57 

Remark. — The  major  and  the  minor  are  consonant  Triads;  all 
the  others  are  dissoncaif  Triads. 

SG.  At  c  the  Triad  differs  from  both  of  the  two  preceding 
ones — it  has  the  minor  Tliird  in  common  with  the  Triad  at  b,  but 
not  the  major  Ft'ffh  J  while  it  differs  from  that  at  rr  in  its  kind 
both  of  llilrd  and  of  Fifth.  The  Triad  at  c,  being  characterized 
by  a  minor  Third  and  Fifth,  may  be  appropriately  named  '"donble- 
minor."  though  it  is  generally  called  '■  diminished."  * 

87.  Let  it  then  be  well  understood  that  the  expressions 
*'  Major  Triad  "  and  '•  Minor  Triad,"  without  any  additional  desig- 


*  The  author  of  this  work  respectfully  submits  that  the  expression 
"  double-minor  "  (?<;^rtf»  the  Triad  concerned,  and  instantly  suggests  its  com- 
position, /.  ('.,  its  kind  of  Third  and  Fifth — which  can  hardly  be  said  of  the 
common  designation  "  diminished."  But  cherished  traditional  names  must 
be  defended  at  all  hazards,  and  so  it  is  said  that  the  Triad  of  which  we  are 
speaking  is  called  '■  diminished  "  because  its  compass  is  "  smaller  by  a  half- 

a 
step  than  the  minor  Triad."     Now  the  altered  Triad  /,  for  instance,  called  by 

d$ 

the  author  of  the  present  work  "  diminished-minor  "  (diminished  Third  and 
minor  Fifth),  is  "smaller  by  a  (chromatic)  half-step  than  a  minor  Triad,"  yet 
does  not  represent  the  so-called  "  diminished  "  Triad.  On  the  other  hand,  in 
this  example  : 


^^H^iS 


the  symphone  at  b  does  represent  the  "  diminished  "  Triad  ;  and  while  it  undoubt- 
edly is  smaller  by  a  half-step  than  the  minor  Triad  at  a,  it  just  as  truly  is 
smaller  by  a  half-step  than  the  innjor  Triad  at  c !  So  tliat  tiie  reason  for 
the  use  of  the  word  "  diminished  "  in  tiiis  connection  is  hardly  satisfactory. 
Should  any  one  object  to  the  proposed  substitution  of  "  double-minor "  fo^ 
"diminished"  through  unwillingness  to  recognize  the  "minor"  Fifth,  the 
objection  is  met  by  the  suggestion  that  the  Triad  whose  name  is  in  dispute 
may  properly  be  called  "  double-minor "  aa  being  composed  of  two  minor 
Thirds,  one  above  the  other. 


58  MODERN     TONALITY. 

nation,  always  imi^ly  the  major  Fifth ;  and  that  when  the  Fifth 
is  other  than  major,  a  second  designation  is  necessary. 

SS.  The  tliree  normal  Triads  exemplified  above  may  be  modi- 
fied by  the  chromatic  alteration  of  their  tones.  At  present  we 
notice,  in  passing,  only  the  raising  of  the  Fifth  in  the  major 
Triad,  whereby  the  Fifth  is  augmented,  rendering  an  additional 
designation  necessary,  to  distinguish  this  Triad  from  the  major 
Triad. 

Fig.  4^3. 


3iE 


89.  We  may  call  the  above  Triad  "  major-augmented  " — thft 
first  designation  (major)  showing  the  kind  of  Third,  the  seconcj 
(augmented)  the  kind  of  Fifth,  as  is  the  case  with  the  names  of 
the  other  altered  Triads. 

Eemaek.— In  this  chapter  only  so  much  is  said  of  the  Triad 
as  seemed  necessary  as  a  preparation  for  the  remaining  chapters. 
To  the  Remark  to  S5,  that  the  only  consonant  Triads  are  th© 
major  and  the  minor,  we  add,  that  every  possible  consonant  com- 
bination in  music,  how  many-voiced  soever,  is  reducible,  in  the  last 
analysis,  to  either  the  one  or  the  other  of  these  simple  three-voiced 
chords.  Of  these  two  Triads  the  major  sAohq  is  strictly  a  primary, 
purely  natural  formation,  as  being  evolved  from  a  fundamen- 
tal by  the  natural  acoustic  law  of  generation.  Thus,  the  piano- 
string  C,  for  instance,  being  made  to  vibrate,  produces  also  its 
Octave  c,  then  successively  g,  c,  e,  g,  etc.,*  in  wldch  series  we 
recognize  the  major  Triad.  This  law  of  acoustics,  by  which  a 
tone  evolves  its  harmonic  over-tones,  has  been  compared,  by  way 


*  Tones  tlius  produced  fonn  what  is  called  "  nature's  harmony,"  the  word 
"  harmony  "  being  used  in  the  narrower  sense  of  an  euphonious  combination 
of  sounds.  Some  writers  find  in  this  natural  harmony  the  starting-point  of 
the  historical  development  of  the  Diatonic  Scale. 


M  0  B  E  R  X     TONALITY.  59 

of  natural  analogy,  to  the  optical  law  whereby  light  is  resolved 
into  the  colors  of  the  prismatic  spectrum. 

QUESTIONS. 

82.  What  is  a  "  Chord  "  '?    (Wheu  is  a  Chord  incomplete  ?  See  Note.) 

83.  Describe  the   Triad 84.  How   are   the    Triads   distiugiiished  ?  ... 

85.  Give  au  example  of  a  major  Triad.  Give  an  example  of  a  minor  Triad. 
What  constitutes  the  diflference  between  the  two?    Wliich  are  the  consonant 

Triads?  (Remfirk.) 86.   Describe  the  double  minor  Triad.  ...87.  What 

kind  of  Fifth  is  always  implied  in  the  major  and  the  minor  Triad?  ... 
88,  89.  What  is  the  "major-augmented"  Triad  ?  F^yevy  \)oss\h\e  consonant 
chord  may  he  reduced,  in  its  simplest  form,  to  what  ?  (In  what  sense  is  the 
major  Triad  alone  a  purely  natural  harmony  ?  Give  an  illustration.  Doess 
nature  furnish  any  analogy  to  the  acoustic  law  of  the  harmonic  over-tones  'i 
(Remark.) 


CHAPTER    XII. 


The  T^ATo  Modes,  or  Keys,  of  Modern  Tonality.— The 
Major  and  the  Minor  Scale. 

90.  The  most  ancient  musical  system  of  which  any  trust- 
worthy accounts  have  come  down  to  us — that  of  the  Greeks — com- 
prised three  distinct  tone-genera,  viz. :  diatonic  (see  9,  ]>.  15), 
chromatic,  and  enharmonic  (see  57,  p.  42).  We  may  form  an  idea 
of  some  kind  as  to  what  their  melodies*  of  the  diafonic  genus  may 
have  been  like,  from  the  structure  of  tlie  different  species  of 
Octaves  or  Scales  illustrated  in  Fig.  7,  p.  17.     But  as  to  the  kind 

*  No  Greek  melodies  have  come  down  to  our  time — at  least,  none  whose 
authenticity  is  satisfactorily  established.  If  is  an  interesting  question, 
whether  harmony,  as  w(;  understand  tlic  expression,  was  known  to  the 
Greeks.     The  general  opinion  among  the  learned  is  that  it  was  not. 


60  MO  D  E  R  X     T  O  XA  L  IT  Y. 

■of  melody  involved  in  the  cliromatic  and  enharmonic  genera  *  we 
can  scarcely  form  an  intelligible  idea,  in  spite  of  the  explanations 
of  Greek  writers  on  the  subject.  Now,  modern  music  recognizes 
the  diatonic  genus  as  the  essential  basis  of  its  tonality,  dividing  it 
into  TWO  hinds,  modes,  or  keys,  viz. :  the  major  and  the  minor, 
and  adopting  for  each,  one  of  the  ancient  diatonic  scales  as  a 
model.  When,  however,  we  say  that  modern  music  is  essentially 
diatonic,  it  is  by  no  means  meant  that  it  is  exclusively  so — on  the 
contrary,  both  the  modes,  or  keys,  are,  in  modern  practice,  tem- 
pered by  the  employment  of  chromatic  tones,  and  allow  the  appli- 
cation of  enharinonics,  in  the  modified  sense  which  has  been  ex- 
plained.    (See  Chap.  VII.) 

91.  We  may  observe  here  that  the  chromatic  element  in  mod- 
ern music  is  to  be  regarded  from  either  a  purely  melodic  or  a  har- 
monic point  of  view,  which  latter,  of  course,  does  not  exclude  the 
former.  Chromatic  tones,  from  the  former  standpoint,  are  em- 
bellishments of  the  melody,  i)assing-notes,  etc.,  not  necessarily 
regarded  in  the  harmony  ;  from  the  latter  standpoint  they  imply 
enrichment  of  the  harmony  as  a  means  of  greater  warmth  and 
intensity  of  expression,  etc.  The  enharmonic  element  is  very 
prominent  in  modern  music,  the  most  striking  and  agreeable  har- 


*  In  the  Greek  system  the  Tetrachord  played  almost  as  important,  a  part 
as  the  Octave  (in  its  broad  sense)  does  in  ours.  The  Tetrachord  is  a  gradual 
succession  of  four  tones  within  the  compass  of  a  minor  Fourth,  as  e-f-g-a, 
which  is  a  Tetrachord  of  the  diatonic  genus.  Tlie  Tetrachords  of  the  chro- 
mntic  genus  consisted  of  a  succession  of  two  half-steps  and  a  minor  Third, 
as  a-ih-bn-d'.  In  the  enharmonic  genus,  the  Tetrachords  progressed  by  what 
.we  may  call — though  not  with  strict  SiCcwMy— quarter -steps,  followed  by  a 
leap  of  a  major  Third,  somewhat  like  this  :  e-eir^f-a.  The  three  genera,  or 
kinds,  ■>az.  :  diatonic,  chromatic,  and  enharmonic,  were  employed  both  sepa- 
rately and  inconjimction  :  though  we  can  hardly  form  any  idea  of  the  practical 
use  of  either  of  the  two  latter  independently.  It  has  been  humorously  ob- 
served that  traces  of  the  enharmonic  genus  are  found  in  the  practice  of 
those  singers  in  whose  vocal  method  the  "portamento  "  plays  so  prominent  a 
part. 


MODERX     TOXALITY.  61 

monic   surprises  being  effected   by  means  of  the  "  enharmonic 
change." 

93.  The  principal  elements  of  our  modern  tonaHty  may,  then, 
be  summed  up  thus :  we  have  only  tioo  modes,  or  keys,  the  one 
major,  the  other  minor ;  each  one  admits  the  employment  of  the 
diatonic  genus,  independently,  as  also  the  mixture  of  the  chromatic 
element  with  the  diatonic:  hence  each  of  the  two  keys  is  provided 
with  its  diatonic  and  its  chromatic-diatonic  scale,  as  models  for 
multiplication  by  transposition,  so  that  either  key  may  be  made 
available  throughout  the  entire  tone-compass.  Moreover,  the 
modern  system  of  tuning  (Equal  Temperament)  brings  witli  it 
the  enharmonic  notatio7i,  whereby  (as  is  explained  in  tlie  science 
of  Harmony)  the  most  manifold  and  intimate  tonal  relationships 
are  easily  established. 

93.  Before  proceeding  to  examine  separately  the  major  and 
the  minor  key,  Ave  must  briefly  consider  the  significance  of  the 
expression  "'  key,"  in  general. 

94,  We  have  said  that  only  two  of  the  ancient  seven  diatonic 
scales  {Fiy.  7,  p.  17)  have  survived  to  our  day,  viz. :  that  of  A  and 
that  of  C,  and  that  these  are  the  model  scales  of  our  modern  tonal- 
ity. A  Scale,  it  m\\  be  borne  in  mind,  represents  the  essential 
tone-material — melodic  and  harmonic — of  a  key*  A  key  consists 
of  a  number  of  different  tones  having  in  their  totality  a  deter- 
minate relationship  to  a  certain  fundamental  tone  called  the  key- 
\one,  or  Tonic  ;  and  a  key  is  either  major  or  minor,  according  to 
the  predominance  of  certain  melodic  or  harmonic  characteristics, 
as  shall  presently  be  explained.  Now,  as  we  shall  soon  see,  the 
scale  of  A  represents  the  minor  key,  and  that  of  6' the  major — 
hence  we  call  the  former  the  model  minor,  the  latter  the  model 
major  scale.  Every  composition  of  modern  times  is  in  one  or  the 
other  of  these  keys,  i.  e.,  draws  its  essential  tone-material  from  a 

*  The  Scale  represents  tlie  key  me.lodieaUy ;  if  we  add  to  the  Scale  its 
appropriate  Triads,  these  represent  the  key  iMrmonically. 


iJ'Z  MODERN     TONxiLITY. 

scale  identical  in  structure  with  that  of  A  or  of  C.  But  a  scale 
may  be  ideutical  in  structure  with  that  of  A  or  C,  and  yet  differ 
from  it  in  pitch — thus  we  talk  of  the  scale  of  B,  for  instance.  The 
scale  of  B  will  be  found,  on  examination,  to  have  the  same  inte- 
rior structure  as  that  of  either  A  or  U:  in  the  former  case  we  have 
the  scale  of  B  minor,  in  the  latter  that  of  B  major  j  and  as  a  scale 
represents  a  kei/,  we  speak  of  the  ke^  of  B  major,  or  of  B  minor. 
Similarly,  we  may  speak  of  the  key  of  I),  major  or  minor,  of  F, 
G,  B\,,  etc.,  etc.,  major  or  minor.  Strictly  speaking,  however, 
there  are,  as  we  have  said,  only  two  really  different  keys,  viz.:  the 
major,  and  the  minor,  the  former  represented  by  the  normal  dia- 
tonic scale  of  C,  the  latter  by  that  of  A  ;  so  that  every  other  key 
than  that  of  C  and  of  A  can  be  but  a  transiwsition,  implying 
change  in  the  pitcli  only,  not  in  the  structure  of  the  scale  of  the 
key.  Hence  it  appears  that  the  expression  "  key  "  has  a  twofold 
sense :  1st,  a  strict  one,  referring  to  interior  structure  only,  os,  when 
we  say  that  a  piece  is  in  the  major  as  distinguished  from  tJie  minor 
key ;  2d,  a  sense  referring  to  difference  of  pitch  also,  as  when  Ave 
speak  of  the  various  keys,  as,  for  instance,  A\^  major,  B  minor,  D 
major,  etc.,  etc.,  meaning  the  various  transjwsitions  of  the  two 
original  model  scales. 

95.  We  must  call  attention,  before  going  further,  also  to  cer- 
tain additional  names  applied  to  the  more  important  tones  of  the 
Diatonic  Scale,  whether  major  or  minor. 

96.  The  1st  degree  of  the  scale  is  the  seat  of  the  key-tone,  or 
Tonic*  The  Tonic,  as  the  base  of  the  tone-ladder,  is  of  the  first 
importance;  it  is  the  source  of  that  unity  and  consistency  with- 
out which  a  piece  of  music  would  not  be  in  any  particolar  key — 
in  other  words,  would  be  but  an  incoherent,  unmeaning  series 
of  sounds,  without  tonal  form.  Hence,  the  close  of  every  mel- 
ody is   fittingly  made    either   on    the   Tonic  or  on   one  of   the 

*  An  exceptioi;  to  tliis  rule  seems  to  be  made  by  C.  F.  Weitzmann,  in  the 
case  of  the  minor  Scale.     See  Note,  p.  78. 


31  0  D  E  R  N     TONALITY.  63 

two  tones  which  with  the  Tonic  form  a  consonant  Triad,  \\z.: 
its  Tuird  (major  or  minor)  and  its  major  Fifth ;  whilst  in  the 
harmony  of  the  close,  consisting  necessarily  of  these  same  tones  iu 
combination,  the  Tonic  itself,  by  way  of  asserting  its  importance, 
its  fundamental  character,  mast  form  the  lowest  tone,  or  bass,  of 
the  chord.* 

97.  The  tones  of  the  scale  which  are  next  in  importance  are : 
that  on  the  Vth  degree,  called  tlie  Dominant,  and  that  on  the  IVth 
degree,  called  the  Subdomiiiant.  Moreover,  the  tone  on  the  de- 
gree next  above  the  Tonic  is  called  the  Supertonic ;  on  the  third 
above,  the  3Iediant  j  and  on  the  sixth  above,  the  Submediant. 

Remark. — The  Triads  of  the  Dominant  and  Subdomitiant,  to- 
gether with  the  Triad  of  the  Tonic,  form  the  character isfic  har- 
monies of  the  key,\  whether  major  or  mmor.  The  Tonic  har- 
mony at  the  close  of  a  composition  most  frequently  takes  imme- 
■(liately  before  it  either  the  harmony  of  the  Doniinant  or  that  of 


*  "In  modern  music,  all  coherence,  botli  of  melody  and  of  harmony,  all 
relationship,  all  principle,  is  involved  in  the  arrangement  of  notes  vrhich  con- 
stitutes a  Key.  This  arransrement  refers  to  any  note  that  may  be  arbitrarily 
chosen  as  the  key-note.  The  key-note  is  in  a  piece  of  music,  to  speak  com- 
paratively, as  the  point  of  sight  is  in  a  perspective  drawing,  whence  all  the 
lines  diverge,  and  which  regulates  the  proportions  of  all  the  objects  in  the 
picture." — Lictureson  Harmony,  by  G.  A.  Macfarren,  London,  1807. 

f  For  instance,  the  major  key  is  characterized  ly  a  wKJor  Triad  on  the 
Tonic,  Subdominant,  and  Dominant.  It  ought  to  be  observed,  however,  that 
the  characteristics  of  the  major  as  distinguished  from  the  minor  key  are  not 
so  constant  as  to  exclude  interchange  to  a  certain  extent.  Thus  the  tiutjor 
key  occasionally  admits  a  minor  Subdominant  Triad— especially  in  the  Plagal 
Cadence ;  whilst  the  minor  key,  though  it  necessarily  excludes  a  major  Snih 


^5^^^eI^1  .  ypt  admits. 


dominant  Triad  in  the  Plagal  Cadence,  e.  g 

IV 

especially  in  its  characteristic  Plagal  Cadence,  a  close  on  the  Tonic  friad  with 

the   Third  raised— m  oth(ir  words,  on  a  vvijur  Tonic  Triad.     Moreover,  the 

minor  key  frequently  borrows  from  the  major  tlie  Dominant  Triad  (major)  ot 

the  latter,  especially  for  Cadences. 


64  MODERN     TONALITY. 

the  Suhdominant,  thus  forming  what  is  called  tlie  final  Cadence. 
If  the  Touic  harmony  is  immediately  preceded  by  that  of  the 
Dominant,  this  is  called  the  Authentic  Cadence  (expressed  thus : 
V-I) ;  if  by  that  of  the  Subdominant,  the  cadence  is  called  Plagal 
(expressed  thus:  IV-I). 

98.  Whenever,  in  a  diatonic  scale,  the  tone  on  the  Vllth  de- 
gree  is  a  lialf-stei)  below  that  on  the  Vlllth  (or  Octave),  thus" 
forming  a  major  Seventh  to  the  Tonic,  it  is  called  the  Leadinrj- 
tone,  ou  account  of  its  natural  general  tendency  to  the  Octave  of 
the  scale,*  i.  e.,  to  the  Tonic,  as  represented  by  its  Octave.  The 
following  are  examples  of  the  most  important  tones  of  the  scale 
of  C. 

Fifj.  44r. 

Tonic.       Subdominant.    Dominant.     Leading-tone. 


-^ 

IV 

t 
•"■ 

V 

VII 

\^        * 

» 

1 • 1 

» 

I  IV  V  VII 

Remaek. — In  marking  the  degrees  of  the  Diatonic  Scale  by 
the  Roman  numerals,  every  degree  is  numbered  exclusively  w/y^mrc? 
from  the  Tonic.  Hence,  as  in  the  above  Figure,  F,  although  a 
Fifth  below  c,  is  marked  IV,  being  the  lower  Octave  of  degree  IV, 
and  G,  though  a  Fourth  below  c,  is  marked  V,  being  the  lower 
Octave  of  degree  V,  and  B,  though  a  Second  below  c,  is  marked 
VII,  being  the  lower  Octave  of  degree  VII. 

QUESTIONS. 

90.  Wliicli  were  the  three  tone-genera,  or  modes  of  melodic  progression,  of 
the  Greek  musical  system  ?  Can  we  form  any  idea  of  the  nature  of  the  an- 
cient Greek  melodies  ?     What  is  the  essential  basis  of  modern  tonality  V    How 

• 

*  This  tendency  is  most  strongly  exhibited  when  the  Seventh  is  contained 
(as  major  Third)  in  the  Authentic  Cadence,  and  especially  in  the  "Dominant 
Seventh-chord,"  as  explained  in  the  study  of  Harmony. 


M  0  D  E  R  X     TO  KA  LITY.  65 

•mviD.y  modes,  OX  keys,  are  there  in  modern  music,  and  whicli  are  they?  Is 
modern  music  exclusively/ diatonic?  Explain  how  it  is  not?.  ...Ol.  From 
what  two  standpoints  may  the  chromatic  element  in  modern  music  be  re- 
garded ?  What  are  chromatic  tones,  from  the  melodic  standpoint  ?  What  do 
they  imply  from  the  harmonic  standpoint  ?  What  do  you  say  of  the 
enharmonic  element  in  modern  music? 92.  Sum  up  the  chief  ele- 
ments of  modern  tonality.  .  .  .04.  We  speak  of  the  key  of  E  winor,  for 
instance,  of  Bi  major,  D  minor,  etc. ;  yet  there  are,  strictly  speaking,  oMy 
tico  keys,  viz. :  the  major  and  the  minor,  represented  by  the  scales  of  C  and 
A,  respectively  ;  every  other  key,  therefore,  than  G  (major)  or  ^1  (minor), 
can  only  be — what  ?  {Ans.  Either  C  or  ^-1  in  another  pitch,  i.  e.,  transposed.) 
The  word  "  key  "  has,  then,  a  two-fold  sense  :  what  is  the  strict  sense  ?  What 
is  the  secondary,  usual  sense  ?. . .  .90.  What  important  tone  of  the  Scale  is 
seated  on  the  1st  degree  ?  What  is  said  of  the  importance  of  this  tone  ?  On 
what  tone  of  the  Scale  does  every  melody  close  ?  How  does  the  Key-ton* 
(Tonic)  of  a  piece  assert  itself  in  the  harmony  of  the  close  ?. . .  .97.  What  ii^ 
the  tone  on  the  Vth  degree  of  the  Scale  called  ?  That  on  the  IVth  ?  That 
on  the  lid  ?  That  on  the  Illd  ?  That  on  the  Vlth  ?  Which  Triads  form  the 
characteristic  harmonies  of  a  key  ?  How  is  the  final  Cadence  formed  ?  When 
is  this  Cadence  called  "  authentic "  ?  When  "  plagal "  ?  (Remark.) .... 
98.  When  is  the  Vllth  degree  of  a  diatonic  scale  called  "Leading-tone"? 


CHAPTER    XIII. 

The  Major  Key.— Transposition  of  the  Model  Major» 

Scale. 

99.  Of  the  two  keys  of  modern  tonality,  that  one  in  whose 
characteristic  harmonies  the  major  Triad  predominates,  is  called 
the  major  hey,  and  is  especially  adapted  for  mnsic  of  a  cheerful, 
brilliant  and  triumphant  character.  Its  -epresentative  scale  is,  as 
has  been  said,  our  old  acquaintance,  the  Diatonic  Scale  of  C. 

100.  In  examining  the  melodic  structure  of  the  model  scale 
of  C,  we  find  that  the  tivo  half -steps  lie  between  degrees  III-IV, 
and  VII- VIII,  respectively,  all  the  other  progressions  being  steps. 


66 


MODERN     TONALITY. 


Fig.  45. 


m 


I      II     III    IV     V     VI  VII  VIII 


101.  Again,  by  adding  the  Tliinl  and  the  Fifth  to  the  Tonic, 
the  Suhdominant,  and  the  Dominant — the  three  most  important 
degrees  and  the  seats  of  the  characteristic  key-harmonies — we  have 
in  each  case  a  major  Triad ;  hence  this  scale  is  called  "major," 
and  serves  as  the  model  scale  for  the  major  key  in  general. 


Fig.  46. 


tfes^E^l^^ 


IV 


102.  But  we  are  by  no  means  confined,  in  the  major  key,  to 
the  pitch  of  the  model  scale — we  may  obtain  varieties  of  pitch  by 
means  of  transposition.  If  now  we  take  each  one  of  the  remain- 
ing eleven  tones,  viz. :  c^  (or  d\f),  d,  d^  (or  c\i),  etc.,  as  a  new  start- 
ing-point or  Tonic,  and  by  proper  use  of  the  chromatic  signs 
reproduce  each  time  a  diatonic  series  exactly  corresponding  in 
structure  to  the  above  model  major  scale  (Fig.  45),  we  shall  obtain 
eleven  additional  major  scales,  each  one  representing  the  major 
key  in  a  different  compass,  or  pitch. 

103.  We  begin  this  reproduction  by  transposing  the  model 
major  scale  of  C  a  chromatic  half-step  above  and  below,  obtaining 
the  following  major  scales  of  C'|  and  C\^. 


Fig.  47. 


Sionatiire. 


I      II    III    IV 


VI  VII  VIII 


f 


MODERN     T  O  y A  L  IT  Y.  67 

Fig.  4S. 

^^  Signature. 


bm     !7g 


^;=w=^^z^g^^^^ 


I      II     III     IV     V     VI  VII  VIII 

104.  In  each  of  the  above  two  transpositions  every  degree  of 
the  scale  is  chromatically  altered.  The  alterations  are  indicated, 
in  the  case  of  every  transposed  scale,  once  for  all,  at  the  beginning 
of  the  staff",  immediately  after  the  Clef,  the  altered  degrees  being 
affected  by  this  Signature,  as  it  is  called,  throughout  the  whole  of 
a  piece,  or  until  the  signature  is  changed.  Thus,  the  major  scale 
of  C!^  has  a  signature  of  seven  sharps,  that  of  C\,,  a  signature  of 
seven  flats,  as  in  the  examples  immediately  above. 

105.  Xow,  in  transposing  a  model  scale  so  as  to  give  all  the 
12  scales  in  a  systematic  order,  according  to  their  relationship,  we 
do  not  begin  Avith  scales  having  the  greatest  number  of  alterations, 
as  above,  but  follow  a  contrary  course.  That  is,  our  first  transpo- 
sition takes  only  one  alteration,  the  next  will  have  but  tivo  {i.  e., 
one  additional  alteration),  and  so  on — we  pass,  in  other  words, 
from  a  given  scale  to  that  one  7nost  nearly  related  to  it  by  commu- 
nity of  tones.  Two  major  scales  are  related  in  the  fl7'st  degree 
when  they  have  the  same  tones  in  common  but  one — in  the  second 
degree,  when  they  have  the  same  except  tivo,  and  so  on  of  the 
third  and  fourth  degrees.  Or,  to  put  it  more  practically,  two 
major  scales  are  related  in  the  first  degree  if  the  Tonic  of  one 
forms  with  tliat  of  the  other  a  major  Fifth,  or  its  inversion,  a 
minor  Fourth.     Thus,  the  major  scale  of  C  is  related  in  the  first 

degree  to  that  of  Gy   ^),  and    to  that  of  F{p-\);  each  of 

these  latter  scales  therefore  takes  htt  one  alteration.  The  scale 
beginning  a  major  Fifth  above,  or  a  minor  Fo^irth  below  G — that 
of  D — will  take  one  additional  alteration,  and  so  will  that  begin- 
ning a  major  Fifth  below,  or  a  minor  Fourth  above  F — the  scale 
of  B7 ;  therefore  the  major  scale  of  D  is  related  to  that  of  C  in  the 


68 


MODERN     TONALITY. 


second  degree,  as  is  also  that  of  B\^,  and  so  on  of  the  other  degrees. 
This  is  ilhistrated  in  the  following  Figure ;  the  model  scale  stands 
in  the  middle,  above  and  below  it  are  two  transpositions  by  major 
Pifths  or  minor  Fourths : 


Fi(j.  49. — Majob  Scale  of  JD. 


$ 


=«*= 


i«»- 


MaJOR  SCAIiE  OF   G. 


-»'        9- 


Model  Ma  joe  Scale  of  G. 


Major  Scale  of  F. 


^ 


s 


Major  Scale  op  J5b. 

_ ^ « — bm^ 


S 


Sicniatiire. 


i 


W 


'm 


106.  As  the  above  figure  shows,  in  transposing  the  scale  by 
the  major  Fifth  above  (or  by  the  minor  Fourth  beloiv),  degree  VII 
of  each  new  scale  is  chromatically  raised;  in  transposing  by  the 
major  Fifth  below  (or  by  the  minor  Fourth  above),  degree  IV  of 
each  new  scale  is  chromatically  lowered.  Now,  this  raising  and 
lowering  can  be  carried  on  with  single  sharps  and  flats  np  to 
the  scales  of  Cj^  and  C\^  (inclusive),  but  7io  farther,  since  in  these 
scales  every  degree  is  already  sharped  or  flatted  (see  Figs.  Ji.7  and 
Jf8).  Plainly,  then,  in  continuing  to  transpose  as  before,  raising 
the  Vllth  or  lowering  the  IVth  degree  of  each  new  scale,  we 
should  have  to  use  the  double-sharp  or  the  double-flat.     Thus, 


MO  D  E  R  X     TO  NA  L  I  TY.  69 

starting  anew  fi'om  C^  and  transposing  by  the  major  Fifth  above, 
or  from  Cj?  and  transposing  by  the  major  Fifth  heloiv,  we  obtain 
in  the  former  case  the  scales  of  G'^  major,  D^  major,  etc.,  and  in 
the  latter  those  of  i^j?  major,  B\f\f  major,  etc.,  written  thus : 

Fig.  50. 

Si-ale  of  Gt  ma,ior.  j(^  Scale  of  Z)f  major. 


f 


ji:3^^^^:^^^^m==;^^E^^3^"^^^ 


^    j{^~ya- 


^    Scale  ot  Kh  nin.1or.  ,       i  bcale  ot  if^r*  mainr.  ,       .      l(,_ 


5^ 

lOT.  The  excessive  chromatic  alteration  in  the  aboTe  scales 
(which  would  increase  with  each  new  transposition)  renders  them 
practically  useless.  We  may,  however,  use  their  key-tones  for 
scales  h)^  enliarmonically  changing  their  notation.  Thus,  G^  being 
written  A\f,  the  scale  of  the  latter  tone  (signature,  4  flats)  is  sub- 
stituted for  that  of  the  former;  D^  being  written  ^[?,  the  scale 
of  the  latter  (signature,  3  flats)  is  used  instead  of  that  of  the 
former ;  and  similarly,  the  scales  of  F\^  and  B\f\}  severally  give 
way  to  those  of  E  (signature,  4  sharps)  and  A  (signature,  3 
sharps). 

108,  The  following  figure  gives  a  table  of  signatures  of 
major  scales  transposed  by  major  Fifths  above  and  below  from  the 
model  major  scale.  Some  of  these  scales  are  inter cha^igeable  at 
pleasure,  viz. :  ^  and  C't?  major;  F'^  and  G\f  major;  C'^  and  D\^ 
major. 

Explanation  of  Fig.  51. — Starting  from  C  we  proceed  to 
the  right  till  avb  come  to  a  signature  containing  a  x,  when  the 
scale  must  be  exchanged  for  that  whose  signature  stands  imme- 
diately underneath :  again,  starting  from  C  at  the  right  we  pro- 
ceed to  the  left  till  we  find  a  signature  with  a  \f\y,  when  the  scale 
Avith  that  signature  gives  way  to  the  scale  whose  signature  stands 
immediately  above. 


70 


MODERN     TONALITY. 


Fiff.  51. 


C  G  D     A     E  B  Ft        Ct         Gt 


Fh 


Cb        Gh       Db 


Ab 


M    Bo  F    C 


Exercises. 


1.  Transpose  the  model  major  Scale  of  C  into  A]  E;  B\  F^\  Gv, ; 

A\^;  D;  E\,;  B\y-  G\^. 

2.  Transpose  the  Chromatic  Scale  of  C  (Fig.  9,  p.  21)   into  Cjj^; 

B;  Ay,  /'jt;  Gfy,  E;  E\^. 


QUESTIONS. 

99.  Which  one  of  the  two  keys  is  called  "  Tnajor "  ?  For  what  kind  of 
music  is  the  major  key  best  adapted  ?  Which  scale  represents  this  key  ?. . . . 
lOO.  What  peculiarity  do  we  find  in  the  melodic  structure  of  the  model 
scale  of  C?  ...101.  Why  is  the  model  scale  of  (7  called  "major"?.... 
102.  How  may  we  have  the  major  key  in  a  'pitch  other  than  that  of  the  scale 
of  C?  How  may  we  obtain  11  additional  major  scales  ?. . . .  104.  In  the  case 
of  a  transposed  scale,  how  are  the  chromatic  alterations  indicated  t  What  is 
this  indication  called'?  What  is  the  effect  of  the  Signature  ?....  105.  In 
transposing  a  model  scale  according  to  a  systematic  order,  how  do  we  begin  ? 
When  are  two  major  '^csAes,  related  in  the  1st  degree?  In  the  2d  degree? 
How  is  the  first  degree  relationship  of  two  major  scales  expressed  more  prac- 
tically?. ..106.  Why  may  we  not  transpose  with  single  sharps  and  flats 
beyond  the  scales  of  Ct  and  Ch  ?  What  follows  from  this  impossibility,  for 
further  transpositions .?....  1 07.  What  is  to  be  said  of  scales  having  in  their 
signature  the  x  or  the  bb  ?  In  what  way,  however,  may  we  utilize  tlieir  key- 
notes ?    Give  examples. . . .  108.  Which  major  scales  are  interchangeable ? 


MODERN     TO  NA  LIT  V.  71 


CHAPTEE     XIY. 

The  Minor  Key. 

109.  The  eminent  modern  musical  theorist,  C.  F.  Weitz- 
mann,  speaking  of  the  minor  key,  says:  "This  key,  which  exhibits 
minor  Triads  as  its  principal  features,  is  of  a  gloomy  and  melan- 
choly character,  and  in  melodic  and  harmonic  respects  the  con- 
stnnt  antithesis  of  the  bright  and  aggressive  major  key,"  etc.,  etc 
{Harmoniesysffm,  Leipzig.^  The  minor  key  is  represented  by 
the  diatonic  scale  of  A,  which  is  therefore  the  ?nodel  minor  scale. 

110.  In  tlie  model  minor  scale  of  A  we  find  that — unlike  th^ 
structure  of  the  major  scale  of  C — the  half-step  lies  between  de- 
grees II-III  and  V-VI. 

Fin.  .->2. 


I      II     III    IV     V     VI  VII  VIII 

Remakk. — As  we  shall  presently  see,  a  iltird  half-step  (between 
degrees  VII  and  VIII)  is  frequently  found  in  the  minor  scale,  im- 
plying, of  course,  chromatic  alteration. 

111.  Again,  by  adding  the  Third  and  the  Fifth  to  the  ToniCy 
ISabdominant,  and  Dominant  of  the  scale  oi  A,  we  obtain  in  each 
case  a  minor  Triad ;  for  which  reason  this  scale  is  called  "minor," 
and  represents  the  minor  key,  in  general. 

Fifj.  /7.'i. 

I  IV  V 


72 


MODERN     TO  NA  L  IT  Y. 


Remark. — The  Triad  of  the  Vth  degree  in  the  minor  key  ia 
often  changed  into  major — a  point  to  which  we  shall  return. 

112.  The  model  scale  of  A  may,  like  that  of  C,  be  transposed, 
any  one  of  the  remaining  eleven  tones  beiug  taken  as  starting- 
point  or  Tonic.  Here  also,  as  in  the  case  of  the  major  scale,  we 
adopt  the  systematic  order  of  transposition,  i.  e.,  by  the  vnajor 
FiftI/,  nhoxe  and  below,  in  following  which  beyond  a  certain  limit 
we  find  that  some  tones  are  not  suitable  for  key-tones,  as  they  are 
ivritten,  inasmuch  as  the  notation  of  their  scales  would  be  too 
complicated.  We  therefore  enharmonically  change  the  names  of 
such  tones,  as  explained  in  the  preceding  chapter. 

113.  The  following  table  of  signatures  exhibits  such  trans- 
positions of  the  model  minor  scale  of  A  as  are  practically  useful, 
the  scales  of  G^  and  A\^  minor  being  interchangeable,  as  are  also 
those  of  D^  and  E  \^  minor : 


Fig.  54:. 


G     D 


114.  It  is  seen  from  iliQ  above  figure  that  the  scale  of  E  minor 
takes  the  signature  of  G  major,  B  minor  that  of  D  major,  F^ 
minor  that  of  A  major,  D  minor  that  of  F  major,  G  jninor  that  of 
B\;  major,  and  so  on.  In  -fact,  the  scale  of  E  minor  has  the  same 
tones  as  that  of  G  major,  the  scale  oi  Dininor  the  same  tones  as  that 
oi  F  major,  and  so  on  of  the  others.  These  and  similar  cases  are 
examples  of  so-called  parallel  or  related  scales  or  keys.  Every 
major  scale  has  its  relative  minor  scale;  the  two  scales  have  tones 
and  signature  in  common,  but  the  minor  scale  starts  from  the 


MODERN     TO  NA  LITY.  73 

minor  TJiird *  below  the  key-note  of  its  relative  major.  Thus,  A 
minor  is  the  relative  minor  scale  to  C  major,  and  conversely,  C 
major  is  the  relative  major  to  A  minor. 

115.  Two  minor  scales  are  related  in  the  first,  second,  or  third 
degree,  precisely  as  two  major  scales  are  under  certain  circum- 
stances, i.  e.,  in  the  jirst  degree  when  the  two  minor  scales  have 
the  same  tones  hut  one;  in  i\\Q  second  w\\q\\  they  have  the  same 
but  hoo,  and  so  on — relationship  of  the  first  degree  existing,  as 
before,  between  two  scales  whose  Tonics  form  a  tnajor  Fifth  or  a 
minor  Fourth.  In  this  way  A  minor  is  related  in  the  first  degree 
to  E  minor  and  to  D  minor,  G  minor  in  the  first  degree  to  D 
minor,  and  in  the  second  to  A  minor,  etc.,  etc. 

116.  When  the  tone  on  the  Vllth  degree  of  the  minor  scale 
is  to  be  used  as  leading-tone  (see  98,  p.  G4)  it  will  be  necessary  to 
<raise  it  by  a  chromatic  half -step,  as  in  the  following  examples : 

Fig.  55. 


t 


jg^EgzJ^u— *=ig 


VII  VII 

117.  It  is,  however,  claimed  by  many  theorists,  that  the 
Seventh  in  the  minor  scale  must  necessarily,  as  Seventh,  be  always 
chromatically  raised — in  other  words,  that  the  minor  is  not  the 
normal  Seventh  in  the  minor  scale.  In  this  little  work  the  con- 
trary doctrine  is  advocated — to  the  understanding  of  which  it  will 
be  necessary  to  consider  the  minor  scale  with  regard  to  its  entire 
harmonic  contents. 

118.  What  we  call  the  Normal  Minor  Scale — as  represented 
by  the  model  scale  of  A,  without  chromatic  alteration — exhibits 
but  three  different  kinds  of  Triads,  viz. :  major,  minor,  and  double- 

*  It  will  sometimes  be  necessary  to  enbarmonically  change  the  notation 
of  this  minor  Tiiird,  as,  for  instance,  to  take  B\i  rather  than  At  as  key-tone  of 
the  relative  minor  scale  to  Ct  major,  the  key  of  At  minor  not  being  in  use. 
4. 


74  MODERN     TONALITY. 

minor,  as  iu  the  following  example  {N.  B.  A  minor  Triad  is  gen- 
erally marked  by  a  small  numeral,  a  major  Triad  by  a  large  one ; 
a  doable-minor  by  a  small  numeral  witii  this  addition  [  °  ],  and 
a  major-augmented  by  a  large  numeral  with  an  accent  [ '  ]  ) : 

Fuj.  aa. — Normal  Minor  Scale. 


-s^    <=^       z^- 


■5^—'^^ — ^  '^ <& ^ 


I     ir  III  IV     V     VI VII 

119.  The  theory  of  the  essetitial  raised  Seventh  in  minor  in- 
volves: 1st,  -d  major — to  the  exclusion  of  a.  minor — Triad  on  the 
Vth  degree  (Dominant) ;  '2d,  a  majoi'-augmented  instead  of  a  major 
Triad  on  the  Illd  degree  (Mediant) ;  and  3d,  a  second  double- 
minor  Triad  on  the  Vllth  degree,  as  in  the  following  example  of 
the  so-called 

Fig.  57' — HARMOJfic  Minor  Scale. 


I     II'  III'  IV    V    VI  vii° 

Remakk. — It  will  be  noticed  that  the  progression  from  the 
Sixth  to  the  raised  Seventh,  in  the  above  scale,  forms  an  aug- 
mented Second. 

120.  The  above  minor  scale  is  called  the  "harmonic,"  be- 
cause, regardless  of  melodic  considerations  (of  which  in  the  next 
paragraph),  it  provides,  both  ascending  and  descending,  for  the 
harmonies  of  the  minor  scale,  specifically  for  a  nmior  Subdominant 
Triad,  and  for  the  assumptively  proper  i)o»u';^««/  harmony,  viz, : 
a  major  Triad  on  the  Vth  degree. 

121.  But  as  the  progression  from  the  Sixth  to  the  raised 
Seventh  (an  augmented  Second),  is  considered  objectionable  on 
melodic  grounds,  the  Sixth  also  is  chromatically  raised,  to  obviate 
this  difficulty,  and  the  minor  scale  appears  in  still  another  form, 


M  0  D  E  R  y     TO  NA  LIT!'.  75 

called  the  "melodic,"  because  its  major  Sixth  (ascending)  and 
minor  Seventh  (descending)  are  adapted  only  to  melodic,  not  to 
harmonic  purposes,  as  above  explained.  In  fact,  the  major  Sixth 
would  make  tlie  iSubdominant  Triad  a  major  one  (which  is  abnor- 
mal in  minor,  as  all  admit),  and  the  minor  Seventh  would  make 
the  Dominant  Triad  a  mi)ior  one,  wliich,  it  is  assumed,  is  inadmis- 
sible; though,  on  the  other  hand,  many  theorists  (with  whom  the 
author  of  this  primer  holds)  maintain  that  the  normal  Triad  of 
the  Dominant  in  the  minor  scale  is  minor,  not  major. 

Fig.  5S. — Melodic  Minor  Scale. 


^^mi 


IV  V 

122.  Thus,  we  have  a  "  harmonic  "  and  a  "  melodic  "  minor 
scale  (the  latter  involving,  in  fact,  two  scales— one  ascending  and 
a  different  one  descending)— hence  much  needless  perplexity  as  to 
the  nature  of  the  minor  key.  The  late  Dr.  A.  B.  Marx,  in  his 
"  Allgemeine  Musiklehre,"  rejects  the  "melodic"  minor  scale, 
though  he  admits  that  the  composer  may  "in  particular  cases  (in 
order  to  make  the  succession  more  soft  and  flowing)  deviate  from 
the  systematic  progression,"  etc. — meaning  by  the  "  systematic 
progression"  the  "harmonic"  scale,  which  he  advocates.  May 
not  we,  with  at  least  equal  consistency,  reject  the  "harmonic" 
scale,  while  admitting  that  the  composer  may  "  in  particular  cases  " 
^say  in  the  case,  e.  g.,  of  the  Seventh,  when  needed  as  leadin<j-t(»u') 
"deviate  from  the"  normal  "progression,"  wliich  we  are  advocat- 
ing ?  Undoubtedly,  any  tone  of  the  minor  scale  fas  well  as  of  the 
major)  may  be  chromatically  altered  at  the  pleasure  of  the  com- 
poser, and  espcciJllly  the  Seventh  may — nay,  7nust — l)e  raised,  wlien 
it  has  to  serve  as  leading-tone,  so  important  and  even  indispensable 
in  a  closing  cadence. 

Vm.  But,  it  will  be  said,  this  is  the  very  reason  why  the  rais- 
ing of  the  Seventh  in  minor  is  essential — because  the  Seventh  is 


76  MODERN     TONALITY. 

the  leading-tone.  Here  is,  however,  a  partial  begging  of  the  ques- 
tion: the  trii2  statement  is,  that  the  Seventh  in  minor  must  be 
be  raised  WHEN  it  has  to  serve  as  leading-tone,  not  BECAUSE 
it  is  leading-tone,  which  it  indisputably  is  not  ALWAYS,  if  the 
practice  of  the  masters  of  harmony  is  any  guide.  In  the  major 
,scale  the  leading-tone  is  indicated  once  for  all  by  the  Signature, 
which  is  never  done  in  the  minor  scale.*  Why  not,  if  the  Seventh 
in  minor  is  always  and  essentially  raised  ?  Tlie  reason  is  evidently 
this,  that  the  Seventh  in  minor  must  be  left  free  (changeable),  so 
as  to  be  able  not  only  to  ascend,  as  leading-tone,  to  the  Octave 
(Tonic),  but  also  to  descend  to  the  Sixth.  Now,  in  the  major 
scale  the  Sixth  (the  normal,  major  Sixth)  can  be  reached  from  the 
.Seventh  in  any  condition — i.  e.,  whether  the  Seventh  be  major  or 
minor;  whereas  in  the  minor  scale  the  normal,  unaltered  Sixth 
can  be  reached  from  the  Seventh  only  on  condition  that  the  latter 
be  minor.  For  example,  in  A  minor,  when  the  Seventh  {g)  is  to 
lead  to  the  Tonic  {a)  in  the  Authentic  Cadence,\  it  should  be 
raised  (f/|±)  :J  l)ut  the  Seventh  does  not  always  lead  upwards  in. 
this  way — it  will  frequently  lead  downward   to  the  Sixth   (/). 


*  An  attempt  was  made  several  years  ago  by  some  music  publishing 
Tiouses  in  Germany  to  introduce  distinctive  signatures  for  the  minor  Scales. 
The  author  of  this  work  has  seen  specimens  of  this  innovation.  The  signa- 
ture of  every  minor  key  contains,  in  addition  to  that  of  its  relative  major 
tey,  a  chromatic  sign  of  raising  on  tlie  Vllth  degree,  inclosed  in  brackets — 


for  instance,  the  scale  of  B  minor :    St»(S)=  .     The  idea  does  not  seem  a 


bad  one,  especially  as  the  bracket  indicates  that  the  sign  of  raising  which  it 
incloses  does  not  essentially  belong  to  the  signature — in  other  words,  that  the 
Vllth  degree  is  cliangeable. 

f  Outside  of  the  Authentic  Cadence  the  Seventh  in  minor  does  not  abso- 
lutely need,  as  every  harmonist  knows,  to  be  raised  in  order  to  move  to  the 
degree  above  (Tonic). 

X  Also,  of  course,  when  it  leads  from  the  Tonic  to  form  the  Third  in  the 
■"  Half-cadence  "  whose  formula  is  1-V. 


MODERX     TOTALITY.  77 

Xow,  it  leads  to  f— properly — not  as  major  Seventh  (gj^)  but  aa 
minor  g),  thus  avoiding  the  augmented  Second. 

124:.  Dr.  Edward  Kriiger  (''  S3'Stem  der  Tonkunst,  Leipzig, 
1866)  says:  "  The  Triads  in  A  minor — the  parallel  of  C,  andthere- 

e     a     h 

fore  the  wormff^  minor  scale  for  us— are   the  following:     c     f    g 

g     c     d'  h  -4    'd    e 

(minor);  e      a     h  (major);   the  Triad  g,  however,  is  less  used 

c      f     g  e 

b 
than  its  chromatically  altered  form  y^ ,  which  serves  for  a  Cadence,"' 

e 
etc.     He  also  maintains  that  there  is  no  good  reason  for  consid- 
ering the  '•  harmonic  "  form  the  ''  correct,  systematic,  proper,  nor- 
mal "  minor  scale. 

125.  "  The  raised  Seventh  in  minor,"  says  the  learned  Arrey 
von  Dommer  in  his  excellent  " Musikalisches  Lexikon,"  "has  its 
siguitieance,  strictly  speaking,  for  the  Cadence  only:  apart  from 
this,  when  there  is  no  occasion  for  the  leading-tone,  the  Seventh. 
(as  well  in  the  Dominant  Triad  as  in  the  Triad  of  the  Illd  and 
that  of  the  Vllth  degree)  is  indeed  frequently  raised,  but  at  least 
as  often  and  with  at  least  as  good  right  left  unaltered.  The  minor 
Seventh  is  characteristic  of  the  minor  scale  ("  leitereigen,"  /.  e., 
'•proper  to  the  scale  ") ;  the  major  Seventh  (leading-tone)  is  intro- 
duced only  for  the  sake  of  a  close,  or  of  modulation,"  etc.,  etc, 
"Moreover"  (he  says  afterwards),  "  the  key  of  the  Dominant  is 
not  major,  but  minor;  *  the  Triad  on  the  Hid  cTegree  with  the 

*  For  instance,  the  key  of  the  Dominant  of  A  minor  is  E  minor,  rehited 
to  A  minor  in  the  first  degree  (see  1 1 5,  p.  73).  And  yet,  if  gt  is  essential  to 
A  minor,  the  related   key — E  minor — cannot  be  represented  (in  its  Tonic 

h 
Triad)  in  A  minor  unless  by  the  major  Triad  gt — which  is  absurd.     The  same 

e 

"  essential "  gim  A  minor  makes  the  Tonic  Triad  of  the  relative  major  key 

{/» 
(P)  appear—on  the  Illd  degree  in  A  minor — as  e  ( ! ).     Thus,  the  minor  Scale^ 


"^S  MO  BE  E  N     TO  N  AL  IT  Y. 

nngmented  Fifth  is  to  be  regarded  only  as  an  altered  Triad  (see 
88,  p.  58),  whilst  with  the  7najor  Fifth  it  has  a  far  higher  sig- 
nificance for  the  scale,  as  being  the  I'Dnic  Triad  of  the  relative 
major  scale  (C).  The  Triad  on  the  Vllth  degree  is  indeed  fre- 
quently changed  into  double-minor  (the  Seventh  being  raised),  but 
appears  often  as  a  major  Triad "  (the  Seventh  remaining  unal- 
tered), etc.,  etc.  "  The  major  Seventh  (in  minor)  is  therefore  not 
absolutely  necessary  except  in  a  Cadence." 

136.  Finally — to  cite  but  one  additional  authority — Mr.  C.  F. 
Weitzmann,  one  of  the  most  learned  and  eminent  of  living  musical 
theorists,  distinctly  acknowledges  and  advocates  the  theory  of  the 
minor  scale  which  we  are  maintaining,  at  least  in  its  essential 
feature — the  unaltered  (minor)  Seventh  as  the  normal  Seventh 
of  the  scale,  with  all  the  harmonic  consequences  which  it  in- 
Tolves.* 

137.  There  is,  in  fact,  no  necessity  for  disputing  as  to  which 
one  of  the  various  forms  of  the  minor  scale  is  the  true  one.  Each 
one  is  good,  for  a  particular  purpose,  but  one  of  them  is  the 
pattern,  the  norm.  It  will  be  conceded  that  the  scale,  in  general, 
yiewed  from  the  inelodic  side,  more  easily  admits  of  variations  than 
when  regarded  from  the  harmonic  side ;   in  practice,  many  chro- 

under  the  influence  of  this  " essential"  raised  Seventh,  appears,  if  we  may 
say  so,  to  disown  its  relations,  or  to  acknowledge  tliem  only  when  they  are 
disguised.  Nay,  .the  aggressiveness  of  this  raised  Seventh  has  in  one  instance 
resulted  in  the  virtual  banishment  of  one  of  the  members  of  the  minor  tone- 
family  (see  Note,  p.  79). 

*  "  The  minor  key,"  says  Weitzmann  ("  Harmoniesystem "),  "  exhibits 
minor  Triads  as  its  principal  features."  He  has,  however,  a  peculiar  theory 
of  the  tonal  succession  in  the  minor  Scale.  According  to  this  theory,  the 
minor  Scale  "  begins  with  the  MftJi  of  the  Tonic  Triad  and  descends  degree 
hy  degree,  leading  by  a  half-step  to  the  Octave  below  the  starting-point,  thus: 
e'  d'  c'  h  a  g  f  e."  The  author  of  this  primer  looks  forward  with  pleasure  to 
the  publication,  one  of  these  days,  of  an  exposition  in  English,  of  Weitz- 
mann's  musical  system,  by  Mr.  Albert  R.  Parsons,  of  New  York,  a  pupil  of 
Weitzmann.  and  thoroughly  qualified  for  the  task. 


MODERN     TONALITY.  79 

matic  tones  (tones  foreign  to  the  scale,  changing-notes,  passing- 
notes,  etc.)  are  introduced  in  melodies  without  necessarily  implying 
a  great  variety  of  accompanying  chords.  Preeminently  fi-om  the 
harmonic  side,  therefore,  the  scale  should  have  a  fixed,  normal 
form,  such  as  in  fact  we  find  in  the  case  of  the  major  scale ;  and 
yet  no  good  reason  can  be  given  why  the  minor  scale  should  be  so 
far  behind  the  major  in  this  respect  as  to  require  explanations 
"which  do  not  explain,  but  which  rather  increase  the  perplexity  of 
students.* 

128.  Let,  then,  the  Scale  of  A  minor  —  with  the  mhior  Sev- 
enth —  be  taken  as  the  normal,  model  scale  of  the  minor  hey,  Avith 

*  For  instance,  Rjchter  ("  Manual  of  Harmony,"  translated  by  J.  P.  Mor- 
gan. New  York,  G.  Scliirmer)  gives  as  the  Triad  of  degree  III  in  minor,  no 
otlier  than  the  major-augmented — quite  consistently  too,  if  the  raised  Seventh 
is  "  essential"  in  minor.  But  then  he  is  obliged  to  admit  that  on  account  of 
the  "  constrained  or  forced  character  of  the  connection  of  this  chord  with 
other  chords  of  the  same  key  " — all  owing,  be  it  observed,  to  the  perversion 
of  the  normal  Mediant  Triad  by  the  "  essential "  raised  Seventh — the  major- 
augmented  Triad  can  hardly  be  recognized  as  Triad  of  the  Hid  degree  in 
minor,  but  is  more  properly  classed  among  the  "  altered  chords  "  of  the  major 
scale !  It  would  seem,  then,  that  the  minor  Scale  has  practically  no  available 
harmony  on  the  Hid  degree  (just  as  some  theorists — among  them  Richter  and 
G.  A.  Macfarren — teach  that  the  Triad  (minor)  of  the  same  degree  in  mijor 
also  is  practically  useless).  This  looks  very  like  adding — harmonically — insult 
to  injury, — the  Mediant  Triad  in  minor  is  first  spoiled  by  the  arbitrary  augmen- 
tation of  its  Fifth,  and  then,  being  found  difficult  to  manage,  is  virtually  bowed 
out  of  the  scale,  as  being  not  wanted  !  Again,  Dr.  G.  A.  Macfarren  ("  Lec- 
tures on  Harmony "),  considering  that  the  augmented  Fifth  in  the  spoiled 
Mediant  Triad  is  dissonant  to  tlie  fundamental,  maintains  that  in  the  1st  inver- 

c' 
sion  of  this  Triad,  for  instance,  (/t  ,  the  Third  {gt)  must  be  omitted.     How 

e 
much  more  intelligible  the  doctrine  that  the  Seventh  in  minor  is  changeable  ; 
hence  that  the  Mediant  Triad  may  he  major-augmented,  but  is  normally  a 
major  Triad,  in  which  condition  it  freely  and  naturally  mingles  with  the  other 
members  of  its  tone-family,  instead  of  being,  like  the  major-augmented  Triad, 
"constrained  or  forced"  so  to  do. 


80  MODERN     TO  NA  L  I  T  Y. 

the  distinct  understanding,  1st,  that  the  "  melodic"  form  (Fig.  58) 
may  be  freely  used  for  melodic  j^urposes,  i.  e.,  without  involving 
the  liarmony — in  other  words,  without  implying  (as  in  the  ascend- 
ing scale  at  least)  an  invariably  major  Triad  on  the  Dominant  and 
Subdominaut;  and  ^f?,  that  as  far  as  the  harmonies  (Triads)  of 
this  scale  are  concerned :  a)  the  Seventh  may  freely  be  chromat- 
ically raised  at  the  will  of  the  composer  (especially  in  a  Cadence), 
yet  in  its  normal  condition  is  minor  (for  instance,  in  A  minor,  (f 
not  g^),  permitting  a  direct  progression  downward  to  the  Sixth 
(for  instance,  g  to  /),  and  giving  a  minor  Triad  on  the  Domina?if 
(representing  the  Tonic  Triad  of  one  of  the  minor  scales  related  to 
it  in  the  first  degree*),  a  major  Triad  on  the  Mediant  (represent- 
ing the  Tonic  Triad  of  the  relative  major  scale),  and  a  major 
Triad  on  the  Vllth  degree ;  whilst  b),  the  Sixth,  likewise  nor- 
mally minor,  giving  a  minor  Subdominant  Triad  (which  is  the 
7'ule  in  minor),  may  also  be  occasionally  raised,  so  as  to  allow  of 
a  direct  progression  upward  to  the  raised  Seventh  (for  instance, 
f^  to  g^,  rather  than  /to  g^),  and  to  give  (exceptionally)  a  major 
Subdominant  Triad,  to  he  used  as  a  passing  chord,  and  never  in  a 
closing  Plagal  Cadence. 

Having  now  reached  the  limits  of  this  little  work,  we  may 
briefly  sum  up  what  has  been  said.  Starting  from  the  definition 
of  a  musical  tone,  we  have  seen  that  modern  music  recognizes 
tioelve  different  tones,  contained  within  the  compass  of  an  Octave^ 
and,  as  arranged  in  the  two  Scales — the  diatonic  and  the  chro- 
matic—affording all  the  tone- gradations  required  for  the  modern 
genus  or  style  of  music,  which  is  essentially  diatonic,  admitting 
the  mixture  of  the  chromatic  element:  that  each  of  the  twelve 
tones  is  repeated  in  successive  perfect  Octaves  above  and  below, 
thus  that  our  tonal  system  is  comprised  in  a  series  of  perfect  Oc- 


*  The  other  related  minor  Scale  is  tonically  represented  in  the  Subdomi- 
nant Triad. 


MODE  EX     TOXALITT.  81 

tares  in  a  higher  and  a  lower  pitch  :  that  the  acquisition  of  this 
series  of  perfect  Octaves,  each  containing  within  its  limits  35  dif- 
ferent tones  as  to  7iame  reduced  to  twelve  different  tones  as  to 
sound,  is  the  result  of  the  tempered  system  of  tuning :  that  the 
diatonic  Octave  or  Scale,  regarded  as  to  its  structure  (composition 
by  steps  and  half-steps  iutermiugled)  is,  in  modern  music,  only 
twofold  in  sjjecies,  severally  representing  the  essential  tone-mate- 
rial of  the  two  modern  diatonic  tone-families — the  major  and  the 
minor  key,  to  either  one  of  which  keys  every  composition  of  our 
times  may  be  assigned ;  and  that  by  means  of  transposition  of  its 
model  scale  each  key  may  be  represented  in  a  various  pitch—' 
whence  the  great  number  of  major  and  minor  keys  or  scales  used* 
in  modern  music;  though  in  fact  all  scales  of  the  same  key — major 
or  minor — are  absolutely  identical  in  interior  structure,  and  differ 
only  in  the  accident  of  pitch.  To  put  it  still  more  concisely — 
modern  tonality  embraces  a  series  of  12  tones,  with  continuous! 
repetitions  (perfect  Octaves)  above  and  below,  a  certain  series  of 
seven  of  these  tones  representing  the  major  key,  while  anothei 
series  of  seven  represents  the  minor  key;  each  key  having  its 
special  fitness  for  purposes  of  musical  expression,  and  being  repro- 
ducible at  pleasure  in  various  degrees  of  pitch.  It  remains  only 
to  add  that  the  musical  student  who  has  mastered  these  first  prin- 
ciples of  the  tonal  system  of  our  time  is  prepared  to  enter  upon 
,  the  fascinating  study  of  Harmony. 

QUESTIONS. 

109.  What  is  the  distinctive  character  of  the  minor  key  ?  Which  Scale 
is  the  model  Scale  of  this  key  ?. .  . .  1 1().  How  does  the  melodic  fitructure  of 
the  minor  Scale  differ  from  that  of  the  major?  ...111.  VViiy  is  the 
"  minor"  Scale  bo  called  ?.  ...  1 12.  What  is  the  systematic  order  followed  in 
transposinjr  the  model  minor  Scale  ?  When  we  find  that  some  tones  are,  as 
uritten,  not  suitable  for  key-tones,  what  do  we  do  ?. . . .  1 14-.  Wliat  are  par- 
aUel  Scales  ?  Give  examples.  What  is  the  difference  between  a  major  Scale 
and  its  relative  minor  Scale '?....  1 15.  When  are  two  minor  Scales  said  to 


82  MODERN     TONALITY. 

be   related    in   the   1st    degree?      When,   in   the  2d?      Give   instances 

llO.  When  the  Seventh  in  the  minor  Scale  is  to  serve  as  leading-tone,  what 
is  necessary  to  be  done?  ...117.  What  doctrine  is  very  commonly  taught 
with  regard  to  the  Seventh  in  the  minor  Scale  ?     What  is  the  doctrine  of  this 

])rinier  on  the  subject  ? 118.  How  many  kinds  of  Triads  are  found  in  the 

Normal  Minor  Scale  ?    What  kind  of  Triads  on  degrees  I,  IV  and  V  ?     What 

kind  on  degrees  III,  VI  and  VII  ?     What  kind  on  degree  11  ? 1 19.  What 

kind  of  Triad  is  involved,  in  the  so-called  "  Harmonic"  minor  Scale,  on  the 
Vth  degree  ?  On  the  Hid  ?  On  the  Vllth  ?  What  do  you  say  of  the  progres 
sion  from  the  Vlth  to  the  Vllth  degree  (or  vice-versa)  in  the  "  harmonic " 

minor  Scale?  (Remark.) 120.  Why  is  the  "  harmonic  "  minor  Scale  so 

called  ? 121.  What  is  the  theory  of  the  "  melodic"  minor  Scale  ?     Why 

is  the  melodic  minor  Scale  not  adapted  to  harmonic  purposes  ?. . .  .  1 23.  What 
reason  is  given  for  the  "  essential  "  raising  of  the  Seventh  in  minor  ?  What 
is  to  be  said  of  this  argument,  and  what  is  the  true  statement  of  the  case  ? 
Is  the  Seventh  in  mmor  always  nsed  as  leading-tone?  What  good  reason  is 
there  for  the  omission  of  the  leading-tone  in  the  signature  in  minor  gen- 
erally ?  Explain  the  difference  between  the  Seventh  in  major  and  the  Sev- 
enth in  minqr,  in  regard  to  connection  with  the  Sixth.  ...  127.  What  is  to 
be  said  of  the  dispute  as  to  which  is  the  proper  form   of  the  minor  Scale  ? 

From  which   side,  particularly,  should   the   Scale  have  a  fixed   form  ? 

128.  Give,  in  a  few  words,  the  theory  of  the  minor  Scale  advocated  in  this 
work.  Sum  up  the  doctrine  of  this  work,  i.  e.,  the  principal  features  of  mod- 
ern tonality. 


APPEIs'DIX    I. 

The  tlieory  of  the  Chromatic  Scale  presented  in  Chap.  IV, 
23  and  23,  though  most  generally  taught^  is  not  universally  ac- 
cepted. Dr.  G.  A.  Macfarren,  lately  elected  Professor  of  Music  in 
the  University  of  Cambridge,  England,  advocates  the  system  of 
Alfred  Day,  according  to  which  the  chromatic  scale  of  C,  for  in- 
stance, is  as  follows,  the  same  descending  as  ascending :  c,  d\^,  d, 
e\^,  &,f,f!^,  g,  a\^,  a,  h\^,  h.  This  system  is  ingeniously  defended  by 
the  professor  in  his  '"Lectures  on  Harmony"  (London,  Longmans 
&  Co.),  to  which  the  reader  is  referred.  Heinrich  Josef  Vincent, 
of  Vienna,  author  of  a  new  musical  theory,  "  Die  Einheit  in  der 
Ton  welt"  (Leipzig,  1862),  gives  a  scheme  of  the  Chromatic  Scale 
almost  identical  with  that  of  Day :  he  starts  from  C,  and,  as  usual, 
transposes  the  scale  by  the  major  Fifth,  gaining  by  each  transpo- 
sitiim  two  additional  names  of  tones — for  instance,  g\^  and  y'^  in  the 
scale  of  F,  and  in  that  of  G,  ((^  and  c^.  In  this  way,  going 
through  all  the  scales,  he  obtanis  all  the  35  different  names  of  the 
12  tones.     The  following  are  illustrations  of  Vincent's  theory  : 

r  c      4      d      r/J      e      f      fi^     g       a^      a       h^      h 

C  \  ^^'^^ 

I     [,11     II     (til  III    IV    ttiV    V     i,VI  VI   l^VII  VII 

(bill) 

(  F     G\,     G      G^    A     B\^    B       c       4      d       e\^      e 

I    1,11  II  #11  III  IV  j|iv    V    [,vi  VI  [,vii  vn 
(b"i) 


F.i 


84  MODERN     TO  NA  LIT  Y. 


G.-l 


g      ai,      a     a!i      b      c'      4     d'       e\      e'      f      f% 
I     1,11    II    #11    III    IV    JiV    V    l,VI    VI    bVII    VII 

(bill) 

The  usual  notation  of  the  Chromatic-diatonic  Scale  is  certainly 
unsatisfactory — there  is  an  inconsistency  in  writing  the  scale  in 
two  ways,  viz.:  ascending,  with  signs  of  raising  exclusively,  and 
descending,  with  signs  of  depression  exclusively.  In  the  key  of 
C  major,  for  instance,  in  descending  from  G  to  the  tone  a  half-step 
lower,  this  tone  would  much  more  frequently  be  written  F''^  than 
Oy.  so,  too,  (hkewise  in  descending  progressions)  G\  would  occur 
as  often  as  A\f,  D^  as  often  as  F\},  and  6'^  as  often  as  jD\^.  Inas- 
much as  one  and  the  same  tone  takes  now  this  name  now  tliMt, 
according  to  circumstances,  the  usual  form  of  the  Chromatic 
Scale,  while  serving  well  enough  for  the  purposes  of  vocal  or  in- 
strumental practice,  is  utterly  worthless  from  a  harmonic  stand- 
point, and,  as  a  basis  of  orthography,  not  only  worthless  but  pos- 
itively hurtful,  as  calculated  to  lead  astray.  Leaving  then  to 
vocalists  and  instrumentalists  the  dual  form  of  this  scale — invari- 
ably ascending  with  signs  of  raising  and  descending  with  those  of 
depression — we  venture  to  suggest  the  scheme  illustrated  below, 
combining  in  a  vnal  form,  for  both  the  major  and  the  miiior  key, 
the  essential  modifications  of  the  primary  tones  by  raising  and  de- 
pression— a  scheme  which,  if  it  does  not  actually  indicate  which 
one  of  a  pair  of  names  is  to  be  used  in  this  or  that  particular  case 
of  modification  of  a  tone,  has,  on  the  other  hand,  at  least  this 
negative  advantage  over  the  usual  dual  form  of  the  Chromatic 
Scale,  that  it  does  not  indicate  a  notation  which  would  be  m  many 
a  case  a  violation  of  musical  orthography. 

Model  Chromatic-diatonic  Scale  in  Majok. 
3         3         4        5      6        7         8         9        10       11       12 


i 


z^z±a^;;=;r3aij!*_-Sg 


I    ti   bii    II  tn  oiii  III  IV  «iv  bv    V   Jv  bvi  VI  Jvi  bvii  VII  VIII 


MODERX     TOXALITT.  85 

Model  Chromatic-diatonic  Scale  in  Minor. 
1         2        3         4  5      6         7         8         9       10    11    13 

I     Ifi   bii    II    to    III  Jiii  IV  «iv  bv    V    $v    VI  $vi  VII  Jfvii  VIII 

111  the  above  scales,  in  wliich  two  tones  identical  in  sound  are 
tied  thus  — ■  (this  being  the  case  in  the  1st  scale — major — with 
each  pair  of  tones  represented  bj  Mack  notes),  the  2d  tone  forms 
^at  once  the  raised  Icey- note  andthe  in  inor  Second  j  the -ith  tone 
the  augmented  Second  and  the  minor  Third,  and  so  on.  Should 
the  laws  of  musical  orthography  require  for  any  of  the  tones  other 
names  than  those  given  in  the  scale,  the  proper  names  may  be 
drawn  from  the  complete  Enharmonic  Scale  (Fig.  33,  p.  40). 

The  examples  above  given  as  models  illustrate  the  Chromatic- 
diatonic  Scales  of  C  major  and  A  minor  only.  It  is  understood 
of  course  that  in  transposing  into  other  keys,  adherence  to  the 
scheme  will  involve  a  diflFerent  notation  of  some  of  the  tones,  in 
accordance  with  the  requirements  of  the  orthography  proper  to 
each  new  key.  In  illustration  of  this,  two  additional  examples 
here  follow: 

Chromatic-diatonic  Scale  of  Ah  major. 


Chromatic-diatonic  Scale  op  Ft  minor. 


The  author  of  this  work,  in  submitting  the  above  forms,  is  far 
from  imagining  for  an  instant  that  he  has  discovered  something 
new,  they  being  in  fact  but  an  application  of  the  Enharmonic 
Scale,  and  suggested,  with  exclusive  reference  to  the  study  of  Har- 
mony, as  a  substitute  for  the  unsatisfactory  Chromatic  Scale  as 


86  MODERN     TONALITY. 

usually  presented.  It  will  be  found  that  Chromatic-diatonic 
Scales  constructed  on  the  models  above  illustrated  atford  excellent 
opportunity  and  material  for  the  study  of  the  important  subject 
of  Intervals. 


APPENDIX   II. 


Musical  Orthography  is  the  doctrine  of  the  proper  spelling,  or 
notation  of  musical  sounds.  If  every  tone  had  its  own  exclusive 
name — for  instance,  if  the  sound  called  C  were  always  called  by 
this  name  and  never  by  any  other,  and  so  on  of  the  other  tones — 
there  would  be  little  or  no  difficulty  in  musical  notation.  But — 
to  keep  to  the  same  example — the  sound  generally  called  C  is  also 
frequently  written  D  double  flat,  and  B  sharp,  according  to  cir- 
cumstances ;  just  as,  in  language,  the  sound  uttered  in  speaking 
the  letter  C  is  written  "C,"  "See,"  or  "Sea"— each  spelling  con- 
veying a  diiferent  meaning.  Now,  a  good  speller  would  not  make 
such  blunders  as  to  write  "C-voyage,"  " see- voyage,"  the  "sea"  of 
a  bishop,  the  letter  "see,"  etc.,  etc.;  and  similarly,  a  good  musical 
speller  would  not  write  down  a  sound  as  C,  when  it  should  be  no- 
tated  B^,  or  D\f\^,  or  vice-versa,  etc. 

Premising  that  the  cause  of  the  varjdng  notation  of  one  and 
the  same  tone  as  to  sound  lies  in  the  Equal  Temper.iment,  where- 
by one  sound  stands  for  three  (as  explained  in  Chapters  VI  and  VII 
of  this  work),  we  would  briefly  point  out  to  beginners  the  funda- 
mental rules  of  musical  spelling,  the  observance  of  which  will  pre- 
serve them  from  the  grosser  blunders  frequently  met  with  in  the 
compositions  of  amateurs.  One  fruitful  occasion  of  a  good  part  of 
these  blunders  is  the  peculiar  structure  of  the  piano-forte  key- 
board, or  rather — to  speak  more  accurately — the  habit  of  attaching 


MODERN     TO  NA  LITY.  87 

to  the  ivliite  keys  what  we  may  call  their  natural  names  only,  and 
of  regarding  only  the  black  X;^^*^  as  " flats "  and  "sharps."  How 
common  this  habit  is,  every  teacher  of  music  well  knows.  It  is 
important,  therefore,  that  the  pupil  should  familiarize  himself 
with  the  contemplation  of  the  key-board  under  the  "enharmonic" 
aspect,  as  illustrated  in  Fig.  33,  p.  -iO,  of  this  work,  frequently 
reminding  himself  that  each  tone,  far  from  having  one  fixed  nanw, 
varies  its  name  according  to  circumstances.  This  may  be  regarded 
as  the  starting-point,  the  fundamental  rule,  par  excellence,  of 
musical  orthography. 

Sup})Osing,  then,  that  a  musical  idea,  conceived  in  a  certain 
key,  is  to  be  reduced  to  notation,  we  may  consider  it,  1st,  as  con- 
sisting exclusively  of  tones  proper  to  the  Diatonic  scale  of  tlie  key;* 
or,  :3d,  as  comprising  some  tones  foreign  to  the  diatonic  scale  of 
the  key. 

The  first  case  offers  but  little  or  no  difficulty,  except,  perhaps, 
in  those  scales  having  a  great  many  chromatically  altered  tones — 
for  instance,  the  scales  of  C^  major,  C\^  major,  Fj^  major,  G^ 
major,  D^  minor,  etc.  The  difficulty  here  is  not  in  the  use  of  the 
appropriate  Chromatic  Signs,  for  these  are  provided  once  for  all, 
in  the  Signature^     What  is  chiefly  necessary  in  this  particular 

*  The  diatonic  scnle  of  a  key — at  least,  of  a  major  key — excludes  all  de- 
rivative, or  cbromatically  altered,  tones,  except  those  indicated  by  the  Signa- 
ture. For  instance,  C  major  includes  only  prhmtry  tones  ;  D  major,  all  the 
primary  tones  excej^t  F  and  C,  which  are  fureign  to  this  scale,  whilst  proper 
to  the  scale  of  C  major,  in  which,  on  the  other  hand,  F%  and  C%  would  be 
foreign  tones.  A^ain  :  the  major  scale  of  (J^  (or  of  C[,)  includes  only  deriv- 
atke,  and  excludes  all  primary  tones.  Thus,  the  tones  proper  to  the  diatonic 
scale  of  C  major  are  foreign  to  that  of  Cj  (or  6'(,)  major,  and  vice-verx//.  In 
the  scale  of  a  minor  key  there  is  one  altered  tone  frequently  occurrinj^,  which, 
though  not  included  in  the  signature,  is  not  to  be  regarded  as  a  fiirdcfu  tone, 
viz.:  the  raised  Sevent/i,  the  Vllfh  degree  in  minor  being  cliangeahle.  See 
Chapter  XIV  of  this  work,  from  122,  p.  75. 

\  Exception  must  be  made  of  the  cliromatic  raising  of  the  Vllth  degree  in 
minor,  when  necessary.     See  the  preceding  note. 


88 


MODERN     TONALITY. 


case,  is,  to  be  careful  to  notate  each  tone  on  its  proper  sfaf -degree 
of  the  given  scale,  not  being  misled  by  the  piano-forte  key-board 
but  following  the  regular  order  of  the  degrees  and  letters,  reading 
them  from  I  to  II,  from  II  to  III,  and  so  on,  as-in  the  following 
table  of  the  keys  most  in  use.  {JY.B.  A  large  capital  signifies  the 
Tonic  of  a  7fiajnr  Scale,  and  a  small  one  that  of  a  minor  Scale.) 

I  (Tonic,  Key-tone,)     II    III    IV    V    VI    VII 


0,  c  (CJ,  4  C»  .    .    D 

E 

F 

G 

A 

B 

D,  D  (DJ,  D[,)  .    .    .    E 

F 

G 

A 

B 

C 

E,  E  {E\^,  Eb)  .     .    .     F 

G 

A 

B 

C 

D 

r,F(Fj,EJi)    .     .    .     G 

A 

B 

C 

D 

E 

G,  G  (Gi  Gj;)     .      .      .      A 

B 

C 

D 

E 

F 

A,  A  {A\f,  Al?)  .    .    .    B 

C 

D 

E 

F 

G 

B,  B  (Bi,,  B[,)   .    .    .    C 

D 

E 

F 

G 

A 

mg  any  one  of  the  above 

lett( 

?rs  u 

nder 

I  as 

Tc 

diatonic  scale — major  or  minor — the  proper  name  for  every  other 
degree  may  be  ascertained  by  following  the  row  of  names,  reading 
to  the  right.  The  addition  of  chromatic  signs  does  not  affect  the 
order  of  the  letters,  i.  e.,  does  not  allow  of  changing  one  letter,  as 
naming  a  degree,  for  another.  Thus,  in  the  key  of  F^  major  th.e 
Seventh  of  the  scale  is  not  written  F,  as  the  key-board  of  the  piano 
might  suggest,  but,  as  the  Table  shows,  B,  of  some  kind — con- 
sequently, in  this  case,  E  sharp.  Again  :  in  the  key  of  C^  major 
the  Tliird  of  the  scale  is  written  E^,  not  F,  and  the  Seventh,  not 
C,  but  B^ ;  in  the  key  of  (rj?  major  the  Fourth  of  the  scale  is  C^y 
not  B;  in  the  key  of  (7[?  major  the  Foiirth  of  the  scale  is  F\^,  not 
3 — and  so  on.  Similarly,  the  leading-tone  (raised  Seventh)  in  the 
scale  of  G^  minor,  wdll  be  ^x,  not  G]  in  the  scale  o^  B^  minor, 
Cx,  not  B,  etc.,  etc.* 


*  Experience  shows  that  details  like  these  are  not  superfluous:  errors  ol 
notation  like  those  pointed  out  are  committed  even  by  professed  musicians. 


MODERN     TO  NA  L  ITT.  89 

To  sum  up  the  above,  the  rule  for  the  1st  case  ma}'  be  briefly 
put  thus :  Xotate  each  tone  ou  the  proper  staff-degree  of  the  scale 
of  the  key,  /.  e.,  according  to  the  relation  of  the  tone  to  the  Tonic 
— the  chromatic  alterations  being  generally  provided  for  by  the 
Signature. 

The  chief  difficulties  in  musical  orthography  occur  in  the  2d 
case,  which  involves  tones  foreign  to  the  diatonic  scale  of  the  keg. 
If,  in  a  given  instance,  a  tone  of  this  kind  is  intended  as  simply  a 
modification  of  the  preceding  tone,  this  general  rule  will  hold 
good,  viz:  keep  the  note  on  the  degree  of  the  tone  of  which  it  is  a 
modification,  and  add  the  sign  of  raising,  or  of  depression,  as  the 
case  may  be,  remembering  that  to  depress  a  tone  already  flatted, 
the  Wrf  is  used,  and  the  x  to  liaise  a  tone  already  sha7jjed.  Thus, 
in  the  following  example,  B  (from  a  Quartet  by  Spohr)  is  merely 
A,  with  three  modifications  of  tones,  and  would  therefore  be  im- 
properly spelled,  if  written  as  at  C. 

A  B  C 

It  should  be  noted,  that  when  the  modified  tone  is  intended  to 
be — what  it  often  really  is — a  kind  of  leading-tone  (progression  by 
a  half-step  rather  than  by  a  stejj),  it  will  form,  with  the  tone  to 
which  it  leads,  a  diatonic  half-step,  this  latter  tone  being  written 
on  the  degree  above,  or  below."*  (N.  B.  This  kind  of  leading-tone  is 
marked  thus  +,  in  the  examples  below.) 

In  the  case  of  a  doubt  whether  a  foreign  tone  is  to  be  written, 
as  a  modification  of  the  preceding  tone,  hence,  whether  on  the 


*  The  idea  of  "leading-tone"  does  not  necessarily  imply  an  exclusively 
ascendinff  i)TogTession.  In  the  resolution  of  the  "Dominant-Seventh-chord," 
the  Seventh,  on  account  of  its  attraction  downward,  to  form  the  Tldrd  in  tho 
following  (Tonic)  harnu)ny,  may  he  considered — in  this  particular  case — as  a 
kind  of  leading-tone. 


90 


MODERN     TO  NA  LIT  Y. 


same  degree,  oi  on  the  degree  next  above  or  leloiv,  the  decision  will 
be  according  to  ivliat  is  intended.  The  pupil  should  ask  himself: 
"Has  this  modification  the  nature  of  a  leading-tone  ?  If  it  has,  is 
the  leadnig  upward,  or  downward  ?  "  In  illustration  of  this  point, 
compare,  in  the  following  examples,  B  with  A,  D  with  C,  and  F 
with  E,  then  the  d\}  in  B  with  the  cj^  in  D  and  F. 

A  B+  C  D+  E  F  + 

-I— I U— 4-fJ^4-,T-J — '>—^ I — 1-^  I I — I 1 ' i-^-. 


If  the  foreign  tone  has  not  the  charactei^  of  leading-tone  as 
above  explained,  the  decision  as  to  its  orthography  will  again  de- 
pend upon  what  is  intended.  An  example  or  two  will  illustrate 
this. 


-9^ — -<S- 


ES=S3S: 


J'l^te 


1^- 


-m 


^-1- 


zSE^EESz 


Where  a  sudden  and  unexpected  change  of  Tcey  is  intended,  it 
will  often  be  necessary  to  make  an  enharmonic  change  of  notation 
in  the  harmony  which  leads  into  the  new  key,  as  indicated  by  the 
ties  in  the  following  illustrations : 


■^J-,-,j=rf 


"m^^^^m 


/  ^ 


It  should  be  understood  that  nothing  more  is  intended  by  this 
Appendix  than  to  give  to  the  tyro  in  composition  some  general 
ideas  on  the  subject  of  musical  orthography,  concerniug  the  rules 
of  which,  by  the  bye,  there  is  not  perfect  agreement  among  theo- 
rists themselves.  At  any  rate,  the  study  of  these  rules  forms, 
strictly  speaking,  part  of  the  doctrine  of  Musical  Composition,  and 
the  practical  application  of  them  is  best  learned  by  carefully 
studying  the  works  of  the  masters  of  musical  art. 


MODERN     TOXALITY.  91 


APPEXDIX    III. 

Heixeich  Josef  Vixcent,  already  alluded  to,  maintains  in 
his  '•  Einheit  der  Tonwelt "  a  peculiar  theory  of  Consonances  and 
Dissonances,  growing  out  of  his  doctrine  of  Intervals,  which  latter 
may  be  summed  up  thus:  In  a  given  key — say  C — the  tones  of 
the  scale  are  invariably  referred  to  and  counted  upward  from  the 
l-ey-tone,  C,  and  give  ahsolute  intervals ;  thus,  g  is  the  absolute  and 
only  Fifth  in  the  scale,  h  the  absolute  and  only  ISeventh,  etc.,  etc. 
Every  other  wa}'  of  establishing  a  relationship  between  two  tones 
of  the  scale  gives  the  casual  or  relative  interval  ("das  zufallige 
Intervall").  Thus,  e.g.,  from  c  to  d,  ahsolute  Second;  but  from  d 
to  c,  casual  Second ;  from  d  upward  to  c',  casual  Seventh  :  from  c 
upward  to/,  absolute  Fourth,  but  from /downward  to  c,  or  from^^ 
upward  to  c\  casual  Fourth,  etc.  The  circumstance  that  b,  for 
instance,  forms  with  g  a  Third,  with  d  a  Sixth,  etc. ,  is  of  but  little 
moment  in  this  system,  the  important  point  being  to  always  regard 
each  tone  only  in  its  relation  to  the  keg-tone,  hence  to  treat  b  as 
simply  and  absolutely  The  Seventh  of  the  Scale. 

Accordingly,  alluding  to  the  doctrine  usually  taught,  that  with 
any  tone  of  the  scale  as  fundamental  the  following  tones  form 
Consonances,  viz:  its  Tidied,  major  or  minor,  its  Fourth  (minor), 
its  Fiffli  (major),  its  Sixth,  major  or  minor,  and  its  Octave — our 
author  remarks :  "  In  these  seven  Consonances  of  the  Thorough- 
Bass  method,  we  see  a  confounding  of  the  absolute  with  the  casual 
inters'al.  The  fundamental  is,  in  Thorough-bass,  any  tone  which 
happens  to  be  the  lower  one.  We  say,  on  the  contrary :  the  major 
or  the  minor  Third,  as  also  the  major  Fifth,  are  consonant  to  I" 


92  MODEBNTO  NA  L  I  T  Y. 

(i.  e.,  to  the  keij-tone).  "We  have,  moreover,  two  imperfect  co?isO' 
nances,  viz  :  II  and  VI,  and  two  neutrals :  IV  and  [7VII.  For  ex- 
ample, in  C,  d  and  a  form  each  an  imperfect  consonance  to  (7, 
Avhilst  /"and  l)\f  are  each  neutral  in  respect  to  6'." 

The  Dissonances,  in  this  system,  are  found  on  the  following 
•degrees  of  the  Scale :  VII  {i.  e.,  major  Seventh,  or  leading-tone, 
invariably  a  dissonance,  the  princiiml  dissonance),  [jll,  j^ll,  jj:IV, 
and  1>VI.* 

'•  Our  Consonances  e  and  g  are  also,  like  the  key-tone,  provided 
each  with  its  leading-tone"    Thus  e  has  its  d'^,  and  g  its/||. 

"  Our  neutral  is  a  tone  which  appears  sometimes  as  a  weak 
•dissonance,  sometimes  as  a  consonance  admitting  of  being  doubled. 
{Note.  A  dissonance  progresses  either  upward  or  downward:  a 
neutral  may  move  both  upioard  and  doumward.)" 

"  Dissonances  are  leading-tones  tending  upwards  or  down- 
wards." 

In  accordance  with  the  above  theory,  there  is  but  one  absolutely 
consonant  Triad  in  a  key,  major  or  minor,  viz:  the  Triad  of  the 
Tonic— 1,  III,  V,  all  the  other  Triads  of  the  key  being,  relatively 
to  that  of  the  Tonic,  either  dissonant  or  imperfectly  consonant. 


*  See  the  sclieme  of  the  Chromatic  Scale  adopted  by  Yioceat,  Appendix  I 
p.  83  of  this  work. 


INDEX. 


CHAPTER  I. 

Tones,  and  their  Notation. 


PAGES 


1.  In  wliat  sense  the  -word  "  Tone  "  is  to  be  understood  . .  .2.  Our  mod- 
ern tonal  system  embraces  twelve  tones,  seven  of  whicli  are  classed 
as  •'  primary.". . .  .3.  How  tones  are  expressed  to  tlie  eye.     The  Staff. 

4  Clefs 6.  Repetitions  of  tones  in  a  higher  and  a  lower  pitcli, 

called  "  Octaves."  Secondary  (broader)  sense  of  the  word  "  Octave." 
The  "  Counter  Octave,"  "  Great  Octave,"  "  Small  Octave,"  etc.,  etc. 
Questions 9 — 14 

CHAPTER  11. 

Scale  and  Key.    The  Diatonic  Scale  in  General. 

7.  The  "Scale."  Its  distinction  as  Diatonic  or  Chromntic.  . .  .8.  "Key." 
...  .9.  Every  key  represented,  in  its  essential  tone-material,  by  its 
Diatonic  Scale.  The  chromatic-diatonic  genus  of  composition.  The 
pure  diatonic  genus 11.  The  piano  key-board  divided  into  "half- 
steps."     The  "  Half-step  "  and  the  "  Step." 13    Exact  definition  of 

the  Diatonic  Scale,  in  general.  Significance  of  the  word  "diatonic." 
...  .13.  Tlie  seven  different  Diatonic  Scales. . .  .14.  The  only  two  of 
the  Diatonic  Scales  used  in  modern  music ....  Questions 14 — 18 

CHAPTER  ITI. 
Modification  or  Chromatic  Alteration  of  Tones. 

16.  Twelve  tones,  counting  the  intermediate  ones,  embraced  within  an 
Octave. . .  .17.  How  the  intermediate  tones  are  "derivatives"  of  the 


94  MODERN     TONALITY. 

PAGES 

primary  tones. . .  .18.  Modification  of  a  tone  by  raising  or  by  depres- 
sion. The  Sharp,  and  Double-sharp.  The  Flat,  and  Double-flat. . . . 
19.  The  Cancel.  What  it  indicates. ..  .20.  Chromatic  Signs,  and 
what  they  effect,  taken  together.  The  chromatic  nomenclature. 
How  one  staff-degree  may  be  the  seat  of  five  different  tones. . . . 
Questions 18—20 

CHAPTER  IV. 

The  Chromatic  Scale. 

21.  What  constitutes  a  Chromatic  Scale.  This  Sca^e  a  Diatonic  Scale, 
with  intermediate  tones. . .  .22.  Usual  manner  of  writing  the  chro- 
matic Scale. . .  .23.  Theory  of  the  usual  notation  of  the  Chromatic 
Scale  ...24.  Two-fold  distinction  of  Half-steps,  and  of  Steps.  The 
diatonic  half-step.  The  chromatic  half-step.  The  use  of  the  chro- 
matic signs  in  writing  a  half-step  not  necessarily  determinative  of  the 
latter  as  chromatic.  Why  the  Scale  generally  called  "  cliromatic  "  is 
more  properly  called  "  chromatic-diatonic". . .  .25.  Two-fold  notation 
of  the  Step.  The  diatonic  notation  the  most  usual.  Composition  of 
the  diatonic  Step. . .  .Exercises. . .  .Questions 20 — 23 

CHAPTER  V. 

Intervals  in  the  Diatonic  Scale. 

26.  Definition  and  full  significance  of  the  term  "Interval." 27.  The 

two  elements  of  the  interval — Denomination  and  Kind.     The  six 

Denominations  of  Intervals  in   the  Scale 28.  Two-fold  Kind  of 

each  Denomination  of  intervals  in  the  Diatonic  Scale.  Difference 
between  a  minor  and  a  major  Interval.  Proof  that  this  difference  is 
specifically  that  of  a  chromatic  half-step 30.  Seconds.  An  "En- 
harmonic change"  not  a  second 31.  Seconds  in  the  Diatonic  Scale 

of  C.  and  distinction  of  them  as  minor  or  major.  Every  Step  {d\&- 
Xomc)ii  major  Second. ..  .32.  The  minor  Second  identical  vsdth  the 
diatonic  JiaJf-step.    Not  every  half-step  a  minor  Second. . .  .33.  Orad- 

v(d,  as  distinjruished  from  skipping  progression 34.  Thirds  in  the 

Diatonic  Scale  of  C ;  which  are  minor,  which  major. . .  .35.  Fourths 
in  the  Diatonic  Scale  of  C ;  with  what  single  exception  all  are  minor. 


MODERN     TO  NA  LITT.  95 

PAOES 

The  minor  Fourth  often  called  "  perfect,"  and  the  major,  "augmented." 
...  .80.  Fifths  in  the  Diatonic  Scale  of  C ;  with  what  single  exception 
all  are  mnjor.     The  minor  Fifth  often  called  "  diminished,"  and  the 

major,  "perfect."     Harmonic  importance  of  the  Fifth 37.  Sixths 

in  the  Diatonic  Scale  of  C ;  wliich  are  minor,  which  major.  . . . 
38.  Sevenths  in  the  Diatonic  Scale  of  C;  with  what  two  exceptions 
all   are   minor. ..  .39.   The   "Half-step   Formula"   of   Intervals.... 

40.  Application  of  this  Formula 41.  The  Half  steps,  in  using  this 

Foniiula,  le.'<s hy  one  than  the  Formula-number  ... 42.  Table  of  the  For- 
mulas of  minor  and  major  intervals,  with  illustrations. . .  .Questions. .  .24 — 33 

CHAPTER  VI. 

The  Equal  Temperamknt. 

43.  The  timing  of  the  intervals,  in  keyed  instruments,  not  mathemati- 
cally correct,  with  one  exception ....  44.  The  12  tones  of  our  modern 
system  selected  from  among  numerous  other  possible  tones  on  the 
ground  of  their  inter-relationship.  The  two  closest  tone-relation- 
ships. . .  .45.  When  two  tones  form  a  perfect  Octave. . .  .46.  When  two 

tones  form  a  perfect  Fifth Easy  means  of  hearing  a  perfect  Octave 

and  Fifth  . .  .47.  A  series  of  repetitions  of  the  same  tone  obtained  by 
multiplyintf  a  tone  by  the  perfect  Octave.  An  endless  series  of  differ- 
ent tones  obtained  by  multiplying  a  tone  by  the  perfect  Fifth .... 
48.  Impossibility  of  obtaining  a  series  of  perfect  Octaves  from  the 
endless  series  of  perfect  Fifths.  .  .  .49.  How  modern  music  deals  with 
the  35  tones,  and  obtains  perfect  Octaves.  The  compromise  by  which 
this  result  is  brought  about.  The  flatting  of  the  Fifths,  in  timing. 
Merging  of  three  tones  into  one.  The  acoustic  impurity  of  the  inter- 
vals in  the  tem])pred  tuning  not  offensive  to  the  ear.  .The  advantages 
gained  by  the  tempered  system — perfect  Octaves  and  the  Enhanuonic 

notation.     Date  of  the  introduction  of  the  Equal  Temperament 

Questions 33—38 

CHAPTER  VII. 
The  Modern  Enharmonic  Scale. 

50.  Illustrations  of  Equal  Temperament,  in  the  Chromatic  Scale.     One 
sound  answering  for  three  sounds  with  different  names.     Analogy  to 


96  MODERN     TO  NA  LIT  V. 


PAGES 


this,  In  spoken  language.  What  constitutes  musical  orthography. 
51.  Enharmonic  Scale,  exhibiting  only  those  tone-modifications  ex- 
pressed by  the  black  keys  of  the  piano-forte. . .  .52.  Why  the  Enhar- 
monic Scale  just  mentioned  is  incomplete.  Comprehensiveness  of  the 
Enharmonic  theory,  whereby  every  tone  (including  those  represented 
by  the  white  keys)  may  be  related  to  the  tone  on  the  staff-degree  next 
above  or  below.     Illustrations.     An  enlarged  Enharmonic  Scale.... 

53.  Why  even  the  enlarged  Enharmonic  Scale  is  still  incomplete. 
The  occasional  need  of  double  raising  or  depression  of  a  tone.  Conse- 
quences of  this,  for  notation.     Tiie  complete  Enharmonic  Scale.... 

54.  Summary  of  the  modern  Enharmonic  theory.  (Why  Gt  alone 
does  not  take  two  additional  names.  See  Note.) ...  .55.  Character- 
istics of  the  Diatonic,  the  Chromatic,  and  the  E)i]iarmoni€  notation, 
abbreviated  for  memorizing. ..  .56.  The  Enharmonic  change. .. . 
57.  The  word  "  enharmonic "  not  used,  in  our  times,  in  its  ancient 
sense.     The  modem  Enharmonic  Scale  simply  a  Chromatic  Scale  with 

a  manifold  notation  for  each  tone. . .  .Questions 38 — 41? 


CHAPTER  VIII. 

MODIFICATIOX  OF   INTERVALS   BY  CHROMATIC   ALTERATION. 

58.  The  modification  of  intervals  implied  in  that  of  tones. . .  .59.  Chro- 
matic alteration  in  the  broad  sense  not  synonymous  with  the  modified^ 
tion,  strictly  speaking,  of  intervals.  Definition  of  Transposition,  as 
applied  to  an  interval.  .  60.  How  the  chromatic  alteration  in  the 
usual  and  strict  sense  differs  from  the  transposition  of  an  interval. 
...  .61.  Extent  of  chromatic  alteration,  and  specific  effects  of  the  lat- 
ter upon  intervals. . .  .62.  How  to  change  a  minor  into  a  major  inter- 
val, and  contrariwise.  In  what  sense  this  chromatic  alteration  effects 
transposition.  Exercise  in  changing  minor  into  major  intervals,  and 
contrariwise  . .  .63.  How  to  diminish  a  minor  interval.  Coincidence 
in  the  formulas  of  the  diminished  Third  and  major-  Second.  How 
these  two  intervals  differ  in  composition.  Diminution  not  applicable 
to  the  Second,  the  minor  Second  being  the  smallest  interval  used. 
Proper  use  of  the  term  "  diminislied,"and  misapplication  of  it  to  the 

minor  Fifth 64.  Augmentation  of  major  intervals. ..  .65.    How 

augmentation  is  practised.     Coincidence  in  the  formulas  of  the  aug- 
mented Second  and  minor  Third.    How  these  two  intervals  differ  in 


MODERN      TONALITY.  97 

PAGES 

composition.  Misapplication  of  tlie  tenn  "  augmented  "  to  the  major 
Fourth ...  .QQ.  Summary  of  the  doctrine  of  this  chapter.  Table  of 
Half-step  Fonnulas  of  all  the  Intervals,  including  the  chromati-cally 
altered.     Exercises.     Questions 43 — 48 


CHAPTER  IX. 

Inteksion  of  Intekvals. 

67.  What  it  is,  to  invert  an  interval.  A  possible  misconception  pointed 
out.... 68.  The  rule  of  inversion.  ..  .69.  How  inversion  changes  an 
interval. . .  .70.  How  the  changes  of  denondnatioii  are  expressed.  . . . 
71.  Enumeration  of  the  changes  of  kind.  Exercise  in  Inversion. 
Questions  49—50 

CHAPTER  X. 

Intekvals  as  Symphones.    Consonances  and  Dissonances. 

73.  The  "Symphone."  Classification  of  two-voiced  sym phones  as  Con- 
sonances or  Dissonances.  Real  significance  of  this  classification.  . . . 
74.  What  a  Consonance  is. . .  .75.  What  a  Dissonance  is.  Resolution 
of  the  Dissonance. . .  .77.  Preparation  of  dissonances. .  .  .78.  Short 
statement  of  the  difference  between  tlie  consonance  and  the  disso- 
nance. . .  .79.  Which  intervals  are  consonances,  and  which  are  disso- 
nances. . .  .80.  Peculiarity  of  some  dissonances.  Which  dissonances 
at  once  strike  the  ear  as  such.... 81.  Dissonances  not  necessarily 
harsh.  Philosopby  of  the  employment  of  dissonances.  Dr.  Macl'ar- 
ren's  remark  on  the  subject Questions 51 — 55 

CHAPTER  XL 

The  Triad,  or  Three-voiced  Chord. 

82.  The  Chord 83.   The  Triatl,  and  its  structure 84.   How  the 

Triads  in  general  are  distinguislied 85.  Illustration.     Which  are 

consfmant  Triads,  and  which  dixsonnnt. . .  .86.  The  "  double-minor  " 
(commonly  called  "diminished")  Triad 87.  The  major  Fifth  al- 


98  MODERN     TONALITY. 

PAGES 

ways  implied  in  the  major  axxA  the  minor  Triad 88,  89.  Chromatic 

alteration  of  Triads.  The  "  major-augmented  "  Triad.  Every  conso- 
nant combination  in  music  reducible  to  a  major  or  minor  Triad. 
The  major  Triad  alone  of  purely  natural  origin.  Illustration,  and 
analogue. . .  .Questions 56 — 59 

CHAPTER  XII. 

The  Two  Modes,  or  Keys,  of  Modern  Tonality.    Tee  Major  and 

THE  Minor  Scale. 

90.  The  three  tone-genera  in  the  ancient  Greek  musical  system.  The 
diatonic  genus  more  intelligible  to  us  than  the  chromatic-  or  the 
enharmonic.  The  diatonic  genus  the  essential  basis  of  modern 
tonality.  Modern  music  not,  however,  exclusively  diatonic,  but  tem- 
pered by  the  chromatic  element. ..  .91.  How  the  chromatic  element 
in  modern  music  is  to  be  regarded.  Chromatic  tones,  from  the  melo- 
dic stand-point  ;  from  the  harmonic.  The  enharmonic  element. . . . 
93.  Summary  of  the  principal  elements  of  modern  tonality.... 
93,94.  Significance  of  the  expression  "key"  in  general.  Only  tioo 
really  different  keys  recognized  in  modern  music,  viz:  the  major  and 
the  minor,  but  many  transposit'ons  of  these  two.  The  expression 
"key"  used,  therefore,  in  a  two-fold  scmse.  . .  .95,  96.  Additional 
names  of  the  more  important  degrees  of  the  Diatonic  Scale.  Tlie 
Tonic.  ...97.  The  Dominant  and  Subdominant.  The  Supertonic, 
Mediant,  and  Submediant.  Characteristic  harmonies  of  the  key. 
The  two-fold  final  Cadence.  The  A>/thentic  Cadence;  the  Plagal. 
, . .  .98.  Tlie  Leading-tone.  Manner  of  marking  the  degrees  of  the 
diatonic  scale  by  Roman  numerals. . . ,  Questions 59 — 65 


CHAPTER  XIIL 

The  Major  Key.    Transposition  of  the  Model  Major  Scale. 

99.  The  Major  Key,  its  character  and  representative  scale.  . .  .100.  Pecu- 
liarities of  the  melodic  structure  of  tlie  model  scale  of  C  . .  . 
101.  Major  Triads  the  characteristic  harmonies  of  the  key  of  C. . . . 
103.  Variations,  in  pitcJi,  of  the  major  key,  obtained  by  transposinc 


MODERN     TONALITY.  99 

PAGES 

its  model  scale 103.  First  transposition  by  a  chromatic  half-step. 

The  major  scales  of  Ct  and  Cb 104.  The  chromatic  alterations  in 

the  last-mentioned   scales.      The   Signature 105.   The  systematic 

order  of  transposition.  Relationship  in  the  first  and  second  degrees 
between  two  major  scales.  106.  Continuation  of  transposition  by 
Fifths  above  or  below,  giving  scales  with  the  x  and  the  bh  in  their 
si^'nature. . .  .107.  Practical  uselessness  of  the  last-mentioned  scales. 
Their  key-tones  rendered  available  for  new  scales  by  enharmonically 

changing  their  notation.     Illustrations 108.  Table  of  signatures 

of  major  scales  transposed  from  the  model  major  scale.  Inter- 
changeable scales.    Exercises,    Questions 65 — 7( 


CHAPTER   XIV. 

The  Mixor  Key. 

109.  Weitzmann's  characterization  of  the  minor  key.  The  representa- 
tive scale  of  this  key. . .  .110.  Melodic  structure  of  the  model  minor 
scale  of  A. . .  .111.  Min»r  Triads  the  characteristic  harmonies  of  tlie 
key  of  A.  Occasional  alteration  of  the  Triad  of  the  Vth  degree. . . . 
112.  Transposition  of  the  model  scale  of  A.  The  same  order  followed 
as  in  transposing  the  model  major  scale.  Enharmonic  change  in  the 
names  of  some  key-tones. ..  .113.  Signatures  of  transposed  minor 
scales 114.  Parallel  scales.  ..  .115.  First  and  second  degree  rela- 
tionship existing  between  two  ndnor  scales...  116.  The  Vllth 
degree  in  minor  raised,  to  serve  as  leadinrj-Ume. . .  .117.  The  doctrine 
that  the  Vllth  degree  in  minor  must  always  be  raised,  not  advocated 
in  this  work. ..  .118.  Only  three  kinds  of  Triads  found  in  the  nor- 
mal minor  scale. . .  .119.  Triads  in  the  "  harmonic"  minor  scale. . . . 
120.  Why  this  scale  is  called  the  "  harmonic". . .  .121.  Rationale  of 

the  "  melodic"  minor  scale 122.  Dr.  A.  B.  Marx's  rejection  of  the 

'•  melodic"  minor  scale  and  advocacy  of  the  "  harmonic."  Our  rejec- 
tion of  the  "  harmonic,"  and  advocacy  of  the  "  normal  "  minor  scale. 
...123.  The  Seventh  in  minor  not  ahoays  used  as  leading-tone. 
Why  the  raised  Seventh  is  never  included  in  the  Signature  of  a 
minor  key...  124.  Dr.  E.  KrQger  on  the  characteristic  Triads  in 
minor. . .  .125.  Arrey  von  Donimer  on  the  same  subject.  . .  .126.  C.  F. 
AVeitzmann's  theory. .  .  .127.   No  need  of  dispute  about  the  minor 


100  31  0  D  E  R  N     TONALITY. 

PAGES 

scale.  Each  different  fonn  good  for  its  purpose,  but  one  of  them  the 
normal  form. ..  .128.  Proposed  theory  of  the  minor  scale.  Brief 
summary  of  the  whole  doctrine  of  this  work. .  .  .Quesiions 71 — h\ 

APPENDIX  1 83 

APPENDIX    II 86 

APPENDIX   III...   91 

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